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Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Formally, a ring is a set endowed with two binary operations called addition and  multiplication such that the ring is an abelian group with respect to the  addition operator, and the multiplication operator is  associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors define rings without requiring a multiplicative identity and instead call the structure defined above a ring with identity. See .)

Whether a ring is commutative has profound implications on its behavior. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields.

Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of $n × n$ real square matrices with $n ≥ 2$, group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology.

The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They later proved useful in other branches of mathematics such as geometry and analysis.

Definition
A ring is a set $R$ equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms
 * 1) $R$ is an abelian group under addition, meaning that:
 * 2) * $(a + b) + c = a + (b + c)$ for all $a, b, c$ in $R$ (that is, $+$ is associative).
 * 3) * $a + b = b + a$ for all $a, b$ in $R$ (that is, $+$ is commutative).
 * 4) * There is an element $0$ in $R$ such that $a + 0 = a$ for all $R$ in $R$ (that is, $0$ is the additive identity).
 * 5) * For each $a$ in $R$ there exists $−a$ in $a$ such that $a + (−a) = 0$ (that is, $−a$ is the additive inverse of $R$).
 * 6) $R$ is a monoid under multiplication, meaning that:
 * 7) * $(a · b) · c = a · (b · c)$ for all $a, b, c$ in $a$ (that is, $⋅$ is associative).
 * 8) * There is an element $1$ in $R$ such that $a · 1 = a$ and $1 · a = a$ for all $R$ in $R$ (that is, $1$ is the multiplicative identity).
 * 9) Multiplication is distributive with respect to addition, meaning that:
 * 10) * $a · (b + c) = (a · b) + (a · c)$ for all $a, b, c$ in $a$ (left distributivity).
 * 11) * $(b + c) · a = (b · a) + (c · a)$ for all $a, b, c$ in $R$ (right distributivity).

In notation, the multiplication symbol $·$ is often omitted, in which case $a · b$ is written as $ab$.

Variations on the definition
In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a "rng" (IPA: ) with a missing "i". For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. As explained in  below, many authors apply the term "ring" without requiring a multiplicative identity.

Although ring addition is commutative, ring multiplication is not required to be commutative: $R$ need not necessarily equal $ba$. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology.

In a ring, multiplicative inverses are not required to exist. A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field.

The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. The proof makes use of the "$1$", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: $ab + cd = cd + ab$.)

There are a few authors who use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative. For these authors, every algebra is a "ring".

Illustration

The most familiar example of a ring is the set of all integers $R$ consisting of the numbers
 * $$\dots,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots$$

The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.

Some properties
Some basic properties of a ring follow immediately from the axioms:
 * The additive identity is unique.
 * The additive inverse of each element is unique.
 * The multiplicative identity is unique.
 * For any element $ab$ in a ring $\Z,$, one has $x0 = 0 = 0x$ (zero is an absorbing element with respect to multiplication) and $(–1)x = –x$.
 * If $0 = 1$ in a ring $x$ (or more generally, $0$ is a unit element), then $R$ has only one element, and is called the zero ring.
 * If a ring $R$ contains the zero ring as a subring, then $R$ itself is the zero ring.
 * The binomial formula holds for any $R$ and $R$ satisfying $xy = yx$.

Example: Integers modulo 4
Equip the set $$\Z /4\Z = \left\{\overline{0}, \overline{1}, \overline{2}, \overline{3}\right\}$$ with the following operations:
 * The sum $$\overline{x} + \overline{y}$$ in $x$ is the remainder when the integer $x + y$ is divided by $4$ (as $x + y$ is always smaller than $8$, this remainder is either $x + y$ or $x + y − 4$). For example, $$\overline{2} + \overline{3} = \overline{1}$$ and $$\overline{3} + \overline{3} = \overline{2}.$$
 * The product $$\overline{x} \cdot \overline{y}$$ in $y$ is the remainder when the integer $\Z/4\Z$ is divided by $4$. For example, $$\overline{2} \cdot \overline{3} = \overline{2}$$ and $$\overline{3} \cdot \overline{3} = \overline{1}.$$

Then $\Z/4\Z$ is a ring: each axiom follows from the corresponding axiom for $xy$ If $\Z/4\Z$ is an integer, the remainder of $\Z.$ when divided by $4$ may be considered as an element of $x$ and this element is often denoted by "$x mod 4$" or $$\overline x,$$ which is consistent with the notation for $0, 1, 2, 3$. The additive inverse of any $$\overline x$$ in $x$ is $$-\overline x=\overline{-x}.$$ For example, $$-\overline{3} = \overline{-3} = \overline{1}.$$

Example: 2-by-2 matrices
The set of 2-by-2 square matrices with entries in a field $\Z/4\Z,$ is
 * $$\operatorname{M}_2(F) = \left\{ \left.\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right|\ a, b, c, d \in F \right\}.$$

With the operations of matrix addition and matrix multiplication, $$\operatorname{M}_2(F)$$ satisfies the above ring axioms. The element $$\left( \begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right)$$ is the multiplicative identity of the ring. If $$A = \left( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right)$$ and $$B = \left( \begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix} \right),$$ then $$AB = \left( \begin{smallmatrix} 0 & 0 \\ 0 & 1 \end{smallmatrix} \right)$$ while $$BA = \left( \begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix} \right);$$ this example shows that the ring is noncommutative.

More generally, for any ring $\Z/4\Z$, commutative or not, and any nonnegative integer $F$, the square matrices of dimension $R$ with entries in $n$ form a ring; see Matrix ring.

Dedekind
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.

Hilbert
The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897. In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring), so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an equivalence). Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if $a3 − 4a + 1 = 0$ then:
 * $$\begin{align}

a^3 &= 4a-1, \\ a^4 &= 4a^2-a, \\ a^5 &= -a^2+16a-4, \\ a^6 &= 16a^2-8a+1, \\ a^7 &= -8a^2+65a-16, \\ \vdots \ & \qquad \vdots \end{align}$$ and so on; in general, $an$ is going to be an integral linear combination of $1$, $a$, and $a2$.

Fraenkel and Noether
The first axiomatic definition of a ring was given by Adolf Fraenkel in 1915, but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse. In 1921, Emmy Noether gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.

Multiplicative identity and the term "ring"
Fraenkel's axioms for a "ring" included that of a multiplicative identity, whereas Noether's did not.

Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a $1$ for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of $1$ in the definition of "ring", especially in advanced books by notable authors such as Artin, Bourbaki, Eisenbud, and Lang. There are also books published as late as 2022 that use the term without the requirement for a $1$. Likewise, the Encyclopedia of Mathematics does not require unit elements in rings. In a research article, the authors often specify which definition of ring they use in the beginning of that article.

Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a $1$, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." Poonen makes the counterargument that the natural notion for rings would be the direct product rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including the empty sequence.

Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention:
 * to include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit", or "ring with 1".
 * to omit a requirement for a multiplicative identity: "rng" or "pseudo-ring", although the latter may be confusing because it also has other meanings.

Commutative rings

 * The prototypical example is the ring of integers with the two operations of addition and multiplication.
 * The rational, real and complex numbers are commutative rings of a type called fields.
 * A unital associative algebra over a commutative ring $n$ is itself a ring as well as an $R$-module. Some examples:
 * The algebra $1$ of polynomials with coefficients in $8)$.
 * The algebra $$RX_1, \dots, X_n$$ of formal power series with coefficients in $7)$.
 * The set of all continuous real-valued functions defined on the real line forms a commutative $R$-algebra. The operations are pointwise addition and multiplication of functions.
 * Let $R$ be a set, and let $R$ be a ring. Then the set of all functions from $R$ to $\R$ forms a ring, which is commutative if $X$ is commutative.

 * The ring of quadratic integers, the integral closure of $R$ in a quadratic extension of $X$ It is a subring of the ring of all algebraic integers.

 * The ring of profinite integers $R$ the (infinite) product of the rings of $R$-adic integers $c$ over all prime numbers $\Z$.
 * The Hecke ring, the ring generated by Hecke operators.
 * If $c$ is a set, then the power set of $c$ becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.

Noncommutative rings

 * For any ring $\Z$ and any natural number $\Q.$, the set of all square $\Z[\zeta_n].$-by-$K$ matrices with entries from $K$, forms a ring with matrix addition and matrix multiplication as operations. For $R[X]$, this matrix ring is isomorphic to $\C$ itself. For $\Z$ (and $\Z$ not the zero ring), this matrix ring is noncommutative.
 * If $n = 4$ is an abelian group, then the endomorphisms of $n = 3$ form a ring, the endomorphism ring $n = 1$ of $n > 1$. The operations in this ring are addition and composition of endomorphisms. More generally, if $\C.$ is a left module over a ring $\widehat\Z,$, then the set of all $p$-linear maps forms a ring, also called the endomorphism ring and denoted by $G$.
 * The endomorphism ring of an elliptic curve. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.
 * If $G$ is a group and $\Z _p$ is a ring, the group ring of $End(G)$ over $p$ is a free module over $S$ having $G$ as basis. Multiplication is defined by the rules that the elements of $EndR(V)$ commute with the elements of $S$ and multiply together as they do in the group $G$.
 * The ring of differential operators (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most Banach algebras are noncommutative.

Non-rings

 * The set of natural numbers $R$ with the usual operations is not a ring, since $n$ is not even a group (not all the elements are invertible with respect to addition – for instance, there is no natural number which can be added to $G$ to get $G$ as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers $n$ The natural numbers (including $G$) form an algebraic structure known as a semiring (which has all of the axioms of a ring excluding that of an additive inverse).
 * Let $n$ be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as convolution: $$(f * g)(x) = \int_{-\infty}^\infty f(y)g(x - y) \, dy.$$ Then $R$ is a rng, but not a ring: the Dirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of $R$.

Products and powers
For each nonnegative integer $R$, given a sequence $$(a_1,\dots,a_n)$$ of $V$ elements of $R$, one can define the product $$P_n = \prod_{i=1}^n a_i$$ recursively: let $G$ and let $3$ for $0$.

As a special case, one can define nonnegative integer powers of an element $R$ of a ring: $0$ and $P_{0} = 1$ for $P_{m} = P_{m−1}a_{m}$. Then $1 ≤ m ≤ n$ for all $a0 = 1$.

Elements in a ring
A left zero divisor of a ring $R$ is an element $R$ in the ring such that there exists a nonzero element $R$ of $R$ such that $an = an−1a$. A right zero divisor is defined similarly.

A nilpotent element is an element $\N$ such that $n ≥ 1$ for some $a^{m+n} = a^{m}a^{n}$. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.

An idempotent $$e$$ is an element such that $m, n ≥ 0$. One example of an idempotent element is a projection in linear algebra.

A unit is an element $(\N, +)$ having a multiplicative inverse; in this case the inverse is unique, and is denoted by $ab = 0$. The set of units of a ring is a group under ring multiplication; this group is denoted by $an = 0$ or $n > 0$ or $e2 = e$. For example, if $\Z.$ is the ring of all square matrices of size $R$ over a field, then $a–1$ consists of the set of all invertible matrices of size $R$, and is called the general linear group.

Subring
A subset $R$ of $n$ is called a subring if any one of the following equivalent conditions holds:
 * the addition and multiplication of $n$ restrict to give operations $R×$ making $R$ a ring with the same multiplicative identity as $a$.
 * $R*$; and for all $R$ in $a$, the elements $b$, $U(R)$, and $R$ are in $a$.
 * $a$ can be equipped with operations making it a ring such that the inclusion map $R×$ is a ring homomorphism.

For example, the ring $R$ of integers is a subring of the field of real numbers and also a subring of the ring of polynomials $n$ (in both cases, $n$ contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers $S$ does not contain the identity element $S × S → S$ and thus does not qualify as a subring of $R$ one could call $R$ a subrng, however.

An intersection of subrings is a subring. Given a subset $S$ of $R$, the smallest subring of $x, y$ containing $S$ is the intersection of all subrings of $xy$ containing $−x$, and it is called the subring generated by $1 ∈ S$.

For a ring $S$, the smallest subring of $S$ is called the characteristic subring of $\Z$. It can be generated through addition of copies of $x + y$ and $S → R$. It is possible that $1$ ($\Z[X]$ times) can be zero. If $\Z$ is the smallest positive integer such that this occurs, then $2\Z$ is called the characteristic of $\Z;$. In some rings, $E$ is never zero for any positive integer $2\Z$, and those rings are said to have characteristic zero.

Given a ring $E$, let $1$ denote the set of all elements $R$ in $R$ such that $E$ commutes with every element in $R$: $−1$ for any $E$ in $R$. Then $n · 1 = 1 + 1 + ... + 1$ is a subring of $R$, called the center of $R$. More generally, given a subset $n$ of $n$, let $n$ be the set of all elements in $R$ that commute with every element in $n$. Then $R$ is a subring of $R$, called the centralizer (or commutant) of $x$. The center is the centralizer of the entire ring $R$. Elements or subsets of the center are said to be central in $x$; they (each individually) generate a subring of the center.

Ideal
Let $R$ be a ring. A left ideal of $y$ is a nonempty subset $R$ of $R$ such that for any $R$ in $X$ and $R$ in $S$, the elements $n · 1$ and $R$ are in $X$. If $S$ denotes the $R$-span of $X$, that is, the set of finite sums
 * $$r_1 x_1 + \cdots + r_n x_n \quad \textrm{such}\;\textrm{that}\; r_i \in R \; \textrm{ and } \; x_i \in I,$$

then $R$ is a left ideal if $1$. Similarly, a right ideal is a subset $R$ such that $1$. A subset $R$ is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of $R$. If $I$ is a subset of $R$, then $Z(R)$ is a left ideal, called the left ideal generated by $x, y$; it is the smallest left ideal containing $I$. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of $r$.

If $R$ is in $rx$, then $xy = yx$ and $Z(R)$ are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by $I$. The principal ideal $x + y$ is written as $RI ⊆ I$. For example, the set of all positive and negative multiples of $IR ⊆ I$ along with $RE$ form an ideal of the integers, and this ideal is generated by the integer $Rx$. In fact, every ideal of the ring of integers is principal.

Like a group, a ring is said to be simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.

Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian.

For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal $R I$ of $R$ is called a prime ideal if for any elements $$x, y\in R$$ we have that $$xy \in P$$ implies either $$x \in P$$ or $$y\in P.$$ Equivalently, $I$ is prime if for any ideals $xR$, $RxR$ we have that $(x)$ implies either $2$ or $0$. This latter formulation illustrates the idea of ideals as generalizations of elements.

Homomorphism
A homomorphism from a ring $2$ to a ring $I$ is a function $I$ from $I$ to $I$ that preserves the ring operations; namely, such that, for all $J$, $IJ ⊆ P$ in $R$ the following identities hold:
 * $$\begin{align}

& f(a+b) = f(a) \ddagger f(b) \\ & f(a\cdot b) = f(a)*f(b) \\ & f(1_R) = 1_S \end{align}$$

If one is working with rngs, then the third condition is dropped.

A ring homomorphism $E$ is said to be an isomorphism if there exists an inverse homomorphism to $R$ (that is, a ring homomorphism that is an inverse function). Any bijective ring homomorphism is a ring isomorphism. Two rings $I ⊆ P$, $J ⊆ P$ are said to be isomorphic if there is an isomorphism between them and in that case one writes $$R \simeq S.$$ A ring homomorphism between the same ring is called an endomorphism, and an isomorphism between the same ring an automorphism.

Examples:
 * The function that maps each integer $E$ to its remainder modulo $(R, +, ⋅)$ (a number in $(S, ‡, ∗)$) is a homomorphism from the ring $E$ to the quotient ring $R$ ("quotient ring" is defined below).
 * If $x$ is a unit element in a ring $R$, then $$R \to R, x \mapsto uxu^{-1}$$ is a ring homomorphism, called an inner automorphism of $x$.
 * Let $P$ be a commutative ring of prime characteristic $R$. Then $a$ is a ring endomorphism of $P$ called the Frobenius homomorphism.
 * The Galois group of a field extension $b$ is the set of all automorphisms of $f$ whose restrictions to $R$ are the identity.
 * For any ring $S$, there are a unique ring homomorphism $R$ and a unique ring homomorphism $R$.
 * An epimorphism (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map $f$ is an epimorphism.
 * An algebra homomorphism from a $f$-algebra to the endomorphism algebra of a vector space over $x$ is called a representation of the algebra.

Given a ring homomorphism $S$, the set of all elements mapped to 0 by $\Z$ is called the kernel of $\Z/4\Z$. The kernel is a two-sided ideal of $u$. The image of $R$, on the other hand, is not always an ideal, but it is always a subring of $R$.

To give a ring homomorphism from a commutative ring $R$ to a ring $p$ with image contained in the center of $R$ is the same as to give a structure of an algebra over $L$ to $n$ (which in particular gives a structure of an $R$-module).

Quotient ring
The notion of quotient ring is analogous to the notion of a quotient group. Given a ring $4$ and a two-sided ideal $\Z \mapsto R$ of $\{0, 1, 2, 3\}$, view $\Z\to\Q$ as subgroup of $x ↦ p$; then the quotient ring $L / K$ is the set of cosets of $k$ together with the operations
 * $$\begin{align}

& (a+I)+(b+I) = (a+b)+I, \\ & (a+I)(b+I) = (ab)+I. \end{align}$$ for all $R → 0$ in $k$. The ring $f : R → S$ is also called a factor ring.

As with a quotient group, there is a canonical homomorphism $(R, +, ⋅)$, given by $(R, +, ⋅)$. It is surjective and satisfies the following universal property: For any ring homomorphism $(R, +)$, invoking the universal property with $R / I$ produces a homomorphism $$\overline{f} : R / \ker f \to S$$ that gives an isomorphism from $a, b$ to the image of $f$.
 * If $R / I$ is a ring homomorphism such that $p : R → R / I$, then there is a unique homomorphism $$\overline{f} : R/I \to S$$ such that $$f = \overline{f} \circ p.$$

Module
The concept of a module over a ring generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring $f$, an $R$-module $f$ is an abelian group equipped with an operation $x ↦ x + I$ (associating an element of $S$ to every pair of an element of $R$ and an element of $A$) that satisfies certain axioms. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all $f : R → S$, $f(I) = 0$ in $A$ and all $f : R → S$, $I = ker f$ in $R$,
 * $A$ is an abelian group under addition.
 * $$\begin{align}

& a(x+y) = ax+ay \\ & (a+b)x = ax+bx \\ & 1x = x \\ & (ab)x = a(bx) \end{align}$$ When the ring is noncommutative these axioms define left modules; right modules are defined similarly by writing $A$ instead of $I$. This is not only a change of notation, as the last axiom of right modules (that is $R / ker f$) becomes $R$, if left multiplication (by ring elements) is used for a right module.

Basic examples of modules are ideals, including the ring itself.

Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the dimension of a vector space). In particular, not all modules have a basis.

The axioms of modules imply that $R / I$, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.

Any ring homomorphism induces a structure of a module: if $R / I$ is a ring homomorphism, then $I$ is a left module over $I$ by the multiplication: $R → R / I$. If $R$ is commutative or if $R × M → M$ is contained in the center of $f$, the ring $R$ is called a $R$-algebra. In particular, every ring is an algebra over the integers.

Direct product
Let $R$ and $M$ be rings. Then the product $a$ can be equipped with the following natural ring structure:
 * $$\begin{align}

& (r_1,s_1) + (r_2,s_2) = (r_1+r_2,s_1+s_2) \\ & (r_1,s_1) \cdot (r_2,s_2)=(r_1\cdot r_2,s_1\cdot s_2) \end{align}$$ for all $b$ in $M$ and $x$ in $R$. The ring $y$ with the above operations of addition and multiplication and the multiplicative identity $x(ab) = (xa)b$ is called the direct product of $M$ with $R$. The same construction also works for an arbitrary family of rings: if $M$ are rings indexed by a set $M$, then $ \prod_{i \in I} R_i$ is a ring with componentwise addition and multiplication.

Let $xa$ be a commutative ring and $$\mathfrak{a}_1, \cdots, \mathfrak{a}_n$$ be ideals such that $$\mathfrak{a}_i +  \mathfrak{a}_j = (1)$$ whenever $(ab)x = b(ax)$. Then the Chinese remainder theorem says there is a canonical ring isomorphism: $$R /{\textstyle \bigcap_{i=1}^{n}{\mathfrak{a}_i}} \simeq \prod_{i=1}^{n}{R/ \mathfrak{a}_i}, \qquad x \bmod {\textstyle \bigcap_{i=1}^{n}\mathfrak{a}_i} \mapsto (x \bmod \mathfrak{a}_1, \ldots, x \bmod \mathfrak{a}_n).$$

A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let $$R_i, 1 \le i \le n$$ be rings, $R_i \to R = \prod R_i$ the inclusions with the images $$\mathfrak{a}_i$$ (in particular $$\mathfrak{a}_i$$ are rings though not subrings). Then $$\mathfrak{a}_i$$ are ideals of $ax$ and $$R = \mathfrak{a}_1 \oplus \cdots \oplus \mathfrak{a}_n, \quad \mathfrak{a}_i \mathfrak{a}_j = 0, i \ne j, \quad \mathfrak{a}_i^2 \subseteq \mathfrak{a}_i$$ as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to $S$. Equivalently, the above can be done through central idempotents. Assume that $R$ has the above decomposition. Then we can write $$1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak{a}_i.$$ By the conditions on $$\mathfrak{a}_i,$$ one has that $R$ are central idempotents and $(−1)x = −x$, $f : R → S$ (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let $$\mathfrak{a}_i = R e_i,$$ which are two-sided ideals. If each $S$ is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to $S$.

An important application of an infinite direct product is the construction of a projective limit of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).

Polynomial ring
Given a symbol $R$ (called a variable) and a commutative ring $R$, the set of polynomials
 * $$R[t] = \left\{ a_n t^n + a_{n-1} t^{n -1} + \dots + a_1 t + a_0 \mid n \ge 0, a_j \in R \right\}$$

forms a commutative ring with the usual addition and multiplication, containing $S$ as a subring. It is called the polynomial ring over $R$. More generally, the set $$R\left[t_1, \ldots, t_n\right]$$ of all polynomials in variables $$t_1, \ldots, t_n$$ forms a commutative ring, containing $$R\left[t_i\right]$$ as subrings.

If $S$ is an integral domain, then $rs = f(r)s$ is also an integral domain; its field of fractions is the field of rational functions. If $R$ is a Noetherian ring, then $f(R)$ is a Noetherian ring. If $S$ is a unique factorization domain, then $R × S$ is a unique factorization domain. Finally, $Ri$ is a field if and only if $r1, r2$ is a principal ideal domain.

Let $$R \subseteq S$$ be commutative rings. Given an element $I$ of $R$, one can consider the ring homomorphism
 * $$R[t] \to S, \quad f \mapsto f(x)$$

(that is, the substitution). If $s1, s2$ and $R × S$, then $(1, 1)$. Because of this, the polynomial $R$ is often also denoted by $i ≠ j$. The image of the map $R$ is denoted by $eiej = 0$; it is the same thing as the subring of $R$ generated by $ei$ and $ei$.

Example: $$k\left[t^2, t^3\right]$$ denotes the image of the homomorphism
 * $$k[x, y] \to k[t], \, f \mapsto f\left(t^2, t^3\right).$$

In other words, it is the subalgebra of $i ≠ j$ generated by $R[t]$ and $R[t]$.

Example: let $R$ be a polynomial in one variable, that is, an element in a polynomial ring $t$. Then $R[t]$ is an element in $R[t]$ and $S = R[t]$ is divisible by $R$ in that ring. The result of substituting zero to $R$ in $x = t$ is $f(t) = f$, the derivative of $R$ at $R$.

The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism $$\phi: R \to S$$ and an element $R$ in $R$ there exists a unique ring homomorphism $$\overline{\phi}: R[t] \to S$$ such that $$\overline{\phi}(t) = x$$ and $$\overline{\phi}$$ restricts to $R$. For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring.

To give an example, let $x$ be the ring of all functions from $S$ to itself; the addition and the multiplication are those of functions. Let $f$ be the identity function. Each $f \mapsto f(x)$ in $S$ defines a constant function, giving rise to the homomorphism $f(t)$. The universal property says that this map extends uniquely to
 * $$R[t] \to S, \quad f \mapsto \overline{f}$$

($R$ maps to $x$) where $$\overline{f}$$ is the polynomial function defined by $f$. The resulting map is injective if and only if $R$ is infinite.

Given a non-constant monic polynomial $h$ in $R[x]$, there exists a ring $h$ containing $f$ such that $x$ is a product of linear factors in $k[t]$.

Let $x$ be an algebraically closed field. The Hilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in $$k\left[t_1, \ldots, t_n\right]$$ and the set of closed subvarieties of $S$. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. Gröbner basis.)

There are some other related constructions. A formal power series ring $$R[\![t]\!]$$ consists of formal power series
 * $$\sum_0^\infty a_i t^i, \quad a_i \in R$$

together with multiplication and addition that mimic those for convergent series. It contains $t2$ as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local (in fact, complete).

Matrix ring and endomorphism ring
Let $ϕ$ be a ring (not necessarily commutative). The set of all square matrices of size $S$ with entries in $R$ forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by $t3$. Given a right $x$-module $r$, the set of all $R$-linear maps from $t$ to itself forms a ring with addition that is of function and multiplication that is of composition of functions; it is called the endomorphism ring of $x$ and is denoted by $f(x + h)$.

As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: $$\operatorname{End}_R(R^n) \simeq \operatorname{M}_n(R).$$ This is a special case of the following fact: If $$f: \oplus_1^n U \to \oplus_1^n U$$ is an $f$-linear map, then $R$ may be written as a matrix with entries $f$ in $R[h]$, resulting in the ring isomorphism:
 * $$\operatorname{End}_R(\oplus_1^n U) \to \operatorname{M}_n(S), \quad f \mapsto (f_{ij}).$$

Any ring homomorphism $f(x + h) – f(x)$ induces $(f(x + h) – f(x)) / h$.

Schur's lemma says that if $S$ is a simple right $R$-module, then $f' (x)$ is a division ring. If $$U = \bigoplus_{i = 1}^r U_i^{\oplus m_i}$$ is a direct sum of $f$-copies of simple $k$-modules $$U_i,$$ then
 * $$\operatorname{End}_R(U) \simeq \prod_{i=1}^r \operatorname{M}_{m_i} (\operatorname{End}_R(U_i)).$$

The Artin–Wedderburn theorem states any semisimple ring (cf. below) is of this form.

A ring $kn$ and the matrix ring $R → S$ over it are Morita equivalent: the category of right modules of $R$ is equivalent to the category of right modules over $R[t]$. In particular, two-sided ideals in $n$ correspond in one-to-one to two-sided ideals in $S[t]$.

Limits and colimits of rings
Let $R$ be a sequence of rings such that $R$ is a subring of $R[t]$ for all $U$. Then the union (or filtered colimit) of $R$ is the ring $$\varinjlim R_i$$ defined as follows: it is the disjoint union of all $U$'s modulo the equivalence relation $Mn(R)$ if and only if $EndR(U)$ in $U$ for sufficiently large $R$.

Examples of colimits:
 * A polynomial ring in infinitely many variables: $$R[t_1, t_2, \cdots] = \varinjlim R[t_1, t_2, \cdots, t_m].$$
 * The algebraic closure of finite fields of the same characteristic $$\overline{\mathbf{F}}_p = \varinjlim \mathbf{F}_{p^m}.$$
 * The field of formal Laurent series over a field $f$: $$k(\!(t)\!) = \varinjlim t^{-m}k[\![t]\!]$$ (it is the field of fractions of the formal power series ring $$k[\![t]\!].$$)
 * The function field of an algebraic variety over a field $fij$ is $$\varinjlim k[U]$$ where the limit runs over all the coordinate rings $S = EndR(U)$ of nonempty open subsets $U$ (more succinctly it is the stalk of the structure sheaf at the generic point.)

Any commutative ring is the colimit of finitely generated subrings.

A projective limit (or a filtered limit) of rings is defined as follows. Suppose we are given a family of rings $R → S$, $Mn(R) → Mn(S)$ running over positive integers, say, and ring homomorphisms $EndR(U)$, $Mn(R)$ such that $Mn(R)$ are all the identities and $Mn(R)$ is $Ri + 1$ whenever $x ~ y$. Then $$\varprojlim R_i$$ is the subring of $$\textstyle \prod R_i$$ consisting of $x = y$ such that $k[U]$ maps to $Ri$ under $i$, $Rj → Ri$.

For an example of a projective limit, see .

Localization
The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring $R$ and a subset $mi$ of $R$, there exists a ring $$R[S^{-1}]$$ together with the ring homomorphism $$R \to R\left[S^{-1}\right]$$ that "inverts" $R$; that is, the homomorphism maps elements in $R$ to unit elements in $$R\left[S^{-1}\right],$$ and, moreover, any ring homomorphism from $R$ that "inverts" $Ri$ uniquely factors through $$R\left[S^{-1}\right].$$ The ring $$R\left[S^{-1}\right]$$ is called the localization of $Ri$ with respect to $i$. For example, if $Ri$ is a commutative ring and $Ri$ an element in $Ri$, then the localization $$R\left[f^{-1}\right]$$ consists of elements of the form $$r/f^n, \, r \in R, \, n \ge 0$$ (to be precise, $$R\left[f^{-1}\right] = R[t]/(tf - 1).$$)

The localization is frequently applied to a commutative ring $i$ with respect to the complement of a prime ideal (or a union of prime ideals) in $k$. In that case $$S = R - \mathfrak{p},$$ one often writes $$R_\mathfrak{p}$$ for $$R\left[S^{-1}\right].$$ $$R_\mathfrak{p}$$ is then a local ring with the maximal ideal $$\mathfrak{p} R_\mathfrak{p}.$$ This is the reason for the terminology "localization". The field of fractions of an integral domain $k$ is the localization of $U$ at the prime ideal zero. If $$\mathfrak{p}$$ is a prime ideal of a commutative ring $R$, then the field of fractions of $$R/\mathfrak{p}$$ is the same as the residue field of the local ring $$R_\mathfrak{p}$$ and is denoted by $$k(\mathfrak{p}).$$

If $S$ is a left $R$-module, then the localization of $S$ with respect to $S$ is given by a change of rings $$M\left[S^{-1}\right] = R\left[S^{-1}\right] \otimes_R M.$$

The most important properties of localization are the following: when $R$ is a commutative ring and $S$ a multiplicatively closed subset
 * $$\mathfrak{p} \mapsto \mathfrak{p}\left[S^{-1}\right]$$ is a bijection between the set of all prime ideals in $R$ disjoint from $S$ and the set of all prime ideals in $$R\left[S^{-1}\right].$$
 * $$R\left[S^{-1}\right] = \varinjlim R\left[f^{-1}\right],$$ $R$ running over elements in $f$ with partial ordering given by divisibility.
 * The localization is exact: $$0 \to M'\left[S^{-1}\right] \to M\left[S^{-1}\right] \to M\left[S^{-1}\right] \to 0$$ is exact over $$R\left[S^{-1}\right]$$ whenever $$0 \to M' \to M \to M \to 0$$ is exact over $R$.
 * Conversely, if $$0 \to M'_\mathfrak{m} \to M_\mathfrak{m} \to M_\mathfrak{m} \to 0$$ is exact for any maximal ideal $$\mathfrak{m},$$ then $$0 \to M' \to M \to M \to 0$$ is exact.
 * A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.)

In category theory, a localization of a category amounts to making some morphisms isomorphisms. An element in a commutative ring $R$ may be thought of as an endomorphism of any $R$-module. Thus, categorically, a localization of $R$ with respect to a subset $R$ of $R$ is a functor from the category of $M$-modules to itself that sends elements of $R$ viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, $M$ then maps to $$R\left[S^{-1}\right]$$ and $S$-modules map to $$R\left[S^{-1}\right]$$-modules.)

Completion
Let $Mp$ be a commutative ring, and let $p$ be an ideal of $M$. The completion of $m$ at $1=sm = 0$ is the projective limit $$\hat{R} = \varprojlim R/I^n;$$ it is a commutative ring. The canonical homomorphisms from $s$ to the quotients $$R/I^n$$ induce a homomorphism $$R \to \hat{R}.$$ The latter homomorphism is injective if $S$ is a Noetherian integral domain and $R$ is a proper ideal, or if $S$ is a Noetherian local ring with maximal ideal $R$, by Krull's intersection theorem. The construction is especially useful when $S$ is a maximal ideal.

The basic example is the completion of $f$ at the principal ideal $j ≥ i$ generated by a prime number $S$; it is called the ring of $R$-adic integers and is denoted $R$ The completion can in this case be constructed also from the $R$-adic absolute value on $R$ The $S$-adic absolute value on $R$ is a map $$x \mapsto |x|$$ from $R$ to $S$ given by $$|n|_p=p^{-v_p(n)}$$ where $$v_p(n)$$ denotes the exponent of $R$ in the prime factorization of a nonzero integer $R$ into prime numbers (we also put $$|0|_p=0$$ and $$|m/n|_p = |m|_p/|n|_p$$). It defines a distance function on $R$ and the completion of $I$ as a metric space is denoted by $R$ It is again a field since the field operations extend to the completion. The subring of $R$ consisting of elements $I$ with $Ri → Ri$ is isomorphic to $R$

Similarly, the formal power series ring $Rk → Rj → Ri$ is the completion of $Rk → Ri$ at $k ≥ j ≥ i$ (see also Hensel's lemma)

A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of excellent ring.

Rings with generators and relations
The most general way to construct a ring is by specifying generators and relations. Let $R$ be a free ring (that is, free algebra over the integers) with the set $I$ of symbols, that is, $R$ consists of polynomials with integral coefficients in noncommuting variables that are elements of $I$. A free ring satisfies the universal property: any function from the set $I$ to a ring $\Z$ factors through $p$ so that $(xn)$ is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.

Now, we can impose relations among symbols in $p$ by taking a quotient. Explicitly, if $\Z_p.$ is a subset of $p$, then the quotient ring of $\Q.$ by the ideal generated by $p$ is called the ring with generators $\Q$ and relations $\Q$. If we used a ring, say, $\R$ as a base ring instead of $p$ then the resulting ring will be over $n$. For example, if $$E = \{ xy - yx \mid x, y \in X \},$$ then the resulting ring will be the usual polynomial ring with coefficients in $\Q$ in variables that are elements of $\Q$ (It is also the same thing as the symmetric algebra over $\Q_p.$ with symbols $\Q_p$.)

In the category-theoretic terms, the formation $$S \mapsto \text{the free ring generated by the set } S$$ is the left adjoint functor of the forgetful functor from the category of rings to Set (and it is often called the free ring functor.)

Let $xj$, $xi$ be algebras over a commutative ring $x$. Then the tensor product of $\Z_p.$-modules $$A \otimes_R B$$ is an $F$-algebra with multiplication characterized by $$(x \otimes u) (y \otimes v) = xy \otimes uv.$$

Domains
A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an "ideal" admits prime factorization, fails to be a PID.

Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra. Let $X$ be a finite-dimensional vector space over a field $F$ and $Rj → Ri$ a linear map with minimal polynomial $X$. Then, since $j ≥ i$ is a unique factorization domain, $X$ factors into powers of distinct irreducible polynomials (that is, prime elements): $$q = p_1^{e_1} \ldots p_s^{e_s}.$$

Letting $$t \cdot v = f(v),$$ we make $R$ a $(p)$-module. The structure theorem then says $F$ is a direct sum of cyclic modules, each of which is isomorphic to the module of the form $$k[t] / \left(p_i^{k_j}\right).$$ Now, if $$p_i(t) = t - \lambda_i,$$ then such a cyclic module (for $X$) has a basis in which the restriction of $E$ is represented by a Jordan matrix. Thus, if, say, $F$ is algebraically closed, then all $F$'s are of the form $|x|p ≤ 1$ and the above decomposition corresponds to the Jordan canonical form of $E$. In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.

The following is a chain of class inclusions that describes the relationship between rings, domains and fields:

Division ring
A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem).

Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.

The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.

A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.

Semisimple rings
A semisimple module is a direct sum of simple modules. A semisimple ring is a ring that is semisimple as a left module (or right module) over itself.

Examples

 * A division ring is semisimple (and simple).
 * For any division ring $X$ and positive integer $E$, the matrix ring $R[{[t]}]$ is semisimple (and simple).
 * For a field $A$ and finite group $\Z,$, the group ring $R[t]$ is semisimple if and only if the characteristic of $A$ does not divide the order of $A$ (Maschke's theorem).
 * Clifford algebras are semisimple.

The Weyl algebra over a field is a simple ring, but it is not semisimple. The same holds for a ring of differential operators in many variables.

Properties
Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.)

For a ring $X$, the following are equivalent:
 * $A$ is semisimple.
 * $X$ is artinian and semiprimitive.
 * $R$ is a finite direct product $ \prod_{i=1}^r \operatorname{M}_{n_i}(D_i) $ where each $(t)$ is a positive integer, and each $F → R$ is a division ring (Artin–Wedderburn theorem).

Semisimplicity is closely related to separability. A unital associative algebra $R$ over a field $R$ is said to be separable if the base extension $$A \otimes_k F$$ is semisimple for every field extension $A$. If $V$ happens to be a field, then this is equivalent to the usual definition in field theory (cf. separable extension.)

Central simple algebra and Brauer group
For a field $k$, a $q$-algebra is central if its center is $q$ and is simple if it is a simple ring. Since the center of a simple $V$-algebra is a field, any simple $V$-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a $pi$-algebra. The matrix ring of size $f$ over a ring $k$ will be denoted by $B$.

The Skolem–Noether theorem states any automorphism of a central simple algebra is inner.

Two central simple algebras $pi$ and $f$ are said to be similar if there are integers $D$ and $k$ such that $$A \otimes_k k_n \approx B \otimes_k k_m.$$ Since $$k_n \otimes_k k_m \simeq k_{nm},$$ the similarity is an equivalence relation. The similarity classes $f : V → V$ with the multiplication $$[A][B] = \left[A \otimes_k B\right]$$ form an abelian group called the Brauer group of $p$ and is denoted by $k[t]$. By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.

For example, $k[t]$ is trivial if $k$ is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). $$\operatorname{Br}(\R)$$ has order 2 (a special case of the theorem of Frobenius). Finally, if $D$ is a nonarchimedean local field (for example, $D$), then $$\operatorname{Br}(k) = \Q /\Z $$ through the invariant map.

Now, if $v$ is a field extension of $D$, then the base extension $$- \otimes_k F$$ induces $t – λi$. Its kernel is denoted by $Mn(D)$. It consists of $kG$ such that $$A \otimes_k F$$ is a matrix ring over $v$ (that is, $D$ is split by $D$.) If the extension is finite and Galois, then $ni$ is canonically isomorphic to $$H^2\left(\operatorname{Gal}(F/k), k^*\right).$$

Azumaya algebras generalize the notion of central simple algebras to a commutative local ring.

Valuation ring
If $D$ is a field, a valuation $n$ is a group homomorphism from the multiplicative group $Di$ to a totally ordered abelian group $k$ such that, for any $F / k$, $Rn$ in $G$ with $[A]$ nonzero, $Br(k)$ The valuation ring of $k$ is the subring of $G$ consisting of zero and all nonzero $R$ such that $Br(k)$.

Examples:
 * The field of formal Laurent series $$k(\!(t)\!)$$ over a field $R$ comes with the valuation $R$ such that $Br(k) → Br(F)$ is the least degree of a nonzero term in $R$; the valuation ring of $A$ is the formal power series ring $$k[\![t]\!].$$
 * More generally, given a field $k$ and a totally ordered abelian group $A$, let $$k(\!(G)\!)$$ be the set of all functions from $k$ to $k$ whose supports (the sets of points at which the functions are nonzero) are well ordered. It is a field with the multiplication given by convolution: $$(f*g)(t) = \sum_{s \in G} f(s)g(t - s).$$ It also comes with the valuation $k$ such that $Br(F / k)$ is the least element in the support of $k$. The subring consisting of elements with finite support is called the group ring of $k$ (which makes sense even if $k$ is not commutative). If $n$ is the ring of integers, then we recover the previous example (by identifying $R$ with the series whose $A$th coefficient is $[A]$.)

Rings with extra structure
A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
 * An associative algebra is a ring that is also a vector space over a field $B$ such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of $n$-by-$m$ matrices over the real field $k$ has dimension $Br(F / k)$ as a real vector space.
 * A ring $k$ is a topological ring if its set of elements $k$ is given a topology which makes the addition map ($$+ : R\times R \to R$$) and the multiplication map $K∗$ to be both continuous as maps between topological spaces (where $f$ inherits the product topology or any other product in the category). For example, $\Q _p$-by-$F$ matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.
 * A λ-ring is a commutative ring $k$ together with operations $g$ that are like $F$th exterior powers:
 * $$\lambda^n(x + y) = \sum_0^n \lambda^i(x) \lambda^{n-i}(y).$$
 * For example, $A$ is a λ-ring with $$\lambda^n(x) = \binom{x}{n},$$ the binomial coefficients. The notion plays a central rule in the algebraic approach to the Riemann–Roch theorem.

 * A totally ordered ring is a ring with a total ordering that is compatible with ring operations.

Some examples of the ubiquity of rings
Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.

Cohomology ring of a topological space
To any topological space $F$ one can associate its integral cohomology ring
 * $$H^*(X,\Z ) = \bigoplus_{i=0}^{\infty} H^i(X,\Z ),$$

a graded ring. There are also homology groups $$H_i(X,\Z )$$ of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a $K$-multilinear form and an $v$-multilinear form to get a ($f + g$)-multilinear form.

The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.

Burnside ring of a group
To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.

Representation ring of a group ring
To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure.

Function field of an irreducible algebraic variety
To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.

Face ring of a simplicial complex
Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.

Category-theoretic description
Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of $G$-modules). The monoid action of a ring $K$ on an abelian group is simply an $v$-module. Essentially, an $K$-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".

Let $v(f + g) ≥ min{v(f), v(g)}.$ be an abelian group and let $v(f) ≥ 0$ be its endomorphism ring (see above). Note that, essentially, $v(f)$ is the set of all morphisms of $f$, where if $k$ is in $v(f)$, and $v$ is in $f(n)$, the following rules may be used to compute $n2$ and $⋅ : R × R → R$:
 * $$\begin{align}

& (f+g)(x) = f(x)+g(x) \\ & (f\cdot g)(x) = f(g(x)), \end{align}$$ where $X × X$ as in $λn: R → R$ is addition in $f$, and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, $k + l$, $(A, +)$ is an abelian group. Furthermore, for every $v$ in $k$, right (or left) multiplication by $G$ gives rise to a morphism of $End(A)$, by right (or left) distributivity. Let $End(A)$. Consider those endomorphisms of $G$, that "factor through" right (or left) multiplication of $k$. In other words, let $End(A)$ be the set of all morphisms $v$ of $f$, having the property that $End(A)$. It was seen that every $G$ in $G$ gives rise to a morphism of $G$: right multiplication by $f$. It is in fact true that this association of any element of $n$, to a morphism of $n$, as a function from $n$ to $f + g$, is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian $n$-group (by $\R$-group, it is meant a group with $R$ being its set of operators). In essence, the most general form of a ring, is the endomorphism group of some abelian $R$-group.

Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.

Generalization
Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.

Rng
A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed.

Nonassociative ring
A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.

Semiring
A semiring (sometimes rig) is obtained by weakening the assumption that $f ⋅ g$ is an abelian group to the assumption that $+$ is a commutative monoid, and adding the axiom that $f(x) + g(x)$ for all a in $n$ (since it no longer follows from the other axioms).

Examples:
 * the non-negative integers $$\{0,1,2,\ldots\}$$ with ordinary addition and multiplication;
 * the tropical semiring.

Ring object in a category
Let $n$ be a category with finite products. Let pt denote a terminal object of $R$ (an empty product). A ring object in $n$ is an object $\Z$ equipped with morphisms $$R \times R\;\stackrel{a}\to\,R$$ (addition), $$R \times R\;\stackrel{m}\to\,R$$ (multiplication), $$\operatorname{pt}\stackrel{0}\to\,R$$ (additive identity), $$R\;\stackrel{i}\to\,R$$ (additive inverse), and $$\operatorname{pt}\stackrel{1}\to\,R$$ (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object $X$ equipped with a factorization of its functor of points $$h_R = \operatorname{Hom}(-,R) : C^{\operatorname{op}} \to \mathbf{Sets}$$ through the category of rings: $$C^{\operatorname{op}} \to \mathbf{Rings} \stackrel{\textrm{forgetful}}\longrightarrow \mathbf{Sets}.$$

Ring scheme
In algebraic geometry, a ring scheme over a base scheme $k$ is a ring object in the category of $l$-schemes. One example is the ring scheme $(R, +, ⋅ )$ over $\Z$, which for any commutative ring $R$ returns the ring $(R, +)$ of $R$-isotypic Witt vectors of length $R$ over $A$.

Ring spectrum
In algebraic topology, a ring spectrum is a spectrum $f$ together with a multiplication $$\mu : X \wedge X \to X$$ and a unit map $(R, +)$ from the sphere spectrum $g$, such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category of symmetric spectra.

Special references

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Historical references

 * Bronshtein, I. N. and Semendyayev, K. A. (2004) Handbook of Mathematics, 4th ed. New York: Springer-Verlag ISBN 3-540-43491-7.
 * History of ring theory at the MacTutor Archive
 * Faith, Carl (1999) Rings and things and a fine array of twentieth century associative algebra. Mathematical Surveys and Monographs, 65. American Mathematical Society ISBN 0-8218-0993-8.
 * Itô, K. editor (1986) "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press.
 * Itô, K. editor (1986) "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press.

title: Ring (mathematics)


text:
Caesar cipher

In cryptography, a Caesar cipher, also known as Caesar's cipher, the shift cipher, Caesar's code, or Caesar shift, is one of the simplest and most widely known encryption techniques. It is a type of substitution cipher in which each letter in the plaintext is replaced by a letter some fixed number of positions down the alphabet. For example, with a left shift of 3, D would be replaced by A, E would become B, and so on. The method is named after Julius Caesar, who used it in his private correspondence.

The encryption step performed by a Caesar cipher is often incorporated as part of more complex schemes, such as the Vigenère cipher, and still has modern application in the ROT13 system. As with all single-alphabet substitution ciphers, the Caesar cipher is easily broken and in modern practice offers essentially no communications security.

Example
The transformation can be represented by aligning two alphabets; the cipher alphabet is the plain alphabet rotated left or right by some number of positions. For instance, here is a Caesar cipher using a left rotation of three places, equivalent to a right shift of 23 (the shift parameter is used as the key):

When encrypting, a person looks up each letter of the message in the "plain" line and writes down the corresponding letter in the "cipher" line. Plaintext: THE QUICK BROWN FOX JUMPS OVER THE LAZY DOG Ciphertext: QEB NRFZH YOLTK CLU GRJMP LSBO QEB IXWV ALD

Deciphering is done in reverse, with a right shift of 3.

The encryption can also be represented using modular arithmetic by first transforming the letters into numbers, according to the scheme, A → 0, B → 1, ..., Z → 25. Encryption of a letter x by a shift n can be described mathematically as,

 * $$E_n(x) = (x + n) \mod {26}.$$

Decryption is performed similarly,

 * $$D_n(x) = (x - n) \mod {26}.$$

(Here, "mod" refers to the modulo operation. The value x is in the range 0 to 25, but if x + n or x − n are not in this range then 26 should be added or subtracted.)

The replacement remains the same throughout the message, so the cipher is classed as a type of monoalphabetic substitution, as opposed to polyalphabetic substitution.

History and usage
The Caesar cipher is named after Julius Caesar, who, according to Suetonius, used it with a shift of three (A becoming D when encrypting, and D becoming A when decrypting) to protect messages of military significance. While Caesar's was the first recorded use of this scheme, other substitution ciphers are known to have been used earlier. "'If he had anything confidential to say, he wrote it in cipher, that is, by so changing the order of the letters of the alphabet, that not a word could be made out. If anyone wishes to decipher these, and get at their meaning, he must substitute the fourth letter of the alphabet, namely D, for A, and so with the others.'" His nephew, Augustus, also used the cipher, but with a right shift of one, and it did not wrap around to the beginning of the alphabet: "'Whenever he wrote in cipher, he wrote B for A, C for B, and the rest of the letters on the same principle, using AA for Z.'" Evidence exists that Julius Caesar also used more complicated systems, and one writer, Aulus Gellius, refers to a (now lost) treatise on his ciphers: "'There is even a rather ingeniously written treatise by the grammarian Probus concerning the secret meaning of letters in the composition of Caesar's epistles.'"

It is unknown how effective the Caesar cipher was at the time; there is no record at that time of any techniques for the solution of simple substitution ciphers. The earliest surviving records date to the 9th-century works of Al-Kindi in the Arab world with the discovery of frequency analysis.

A piece of text encrypted in a Hebrew version of the Caesar cipher is sometimes found on the back of Jewish mezuzah scrolls. When each letter is replaced with the letter before it in the Hebrew alphabet the text translates as "YHWH, our God, YHWH", a quotation from the main part of the scroll.

In the 19th century, the personal advertisements section in newspapers would sometimes be used to exchange messages encrypted using simple cipher schemes. David Kahn (1967) describes instances of lovers engaging in secret communications enciphered using the Caesar cipher in The Times. Even as late as 1915, the Caesar cipher was in use: the Russian army employed it as a replacement for more complicated ciphers which had proved to be too difficult for their troops to master; German and Austrian cryptanalysts had little difficulty in decrypting their messages. Caesar ciphers can be found today in children's toys such as secret decoder rings. A Caesar shift of thirteen is also performed in the ROT13 algorithm, a simple method of obfuscating text widely found on Usenet and used to obscure text (such as joke punchlines and story spoilers), but not seriously used as a method of encryption.

The Vigenère cipher uses a Caesar cipher with a different shift at each position in the text; the value of the shift is defined using a repeating keyword. If the keyword is as long as the message, is chosen at random, never becomes known to anyone else, and is never reused, this is the one-time pad cipher, proven unbreakable. However the problems involved in using a random key as long as the message make the one-time pad difficult to use in practice. Keywords shorter than the message (e.g., "Complete Victory" used by the Confederacy during the American Civil War), introduce a cyclic pattern that might be detected with a statistically advanced version of frequency analysis.

In April 2006, fugitive Mafia boss Bernardo Provenzano was captured in Sicily partly because some of his messages, clumsily written in a variation of the Caesar cipher, were broken. Provenzano's cipher used numbers, so that "A" would be written as "4", "B" as "5", and so on.

In 2011, Rajib Karim was convicted in the United Kingdom of "terrorism offences" after using the Caesar cipher to communicate with Bangladeshi Islamic activists discussing plots to blow up British Airways planes or disrupt their IT networks. Although the parties had access to far better encryption techniques (Karim himself used PGP for data storage on computer disks), they chose to use their own scheme (implemented in Microsoft Excel), rejecting a more sophisticated code program called Mujahedeen Secrets "because 'kaffirs', or non-believers, know about it, so it must be less secure".

Breaking the cipher
The Caesar cipher can be easily broken even in a ciphertext-only scenario. Since there are only a limited number of possible shifts (25 in English), an attacker can mount a brute force attack by deciphering the message, or part of it, using each possible shift. The correct description will be the one which makes sense as English text. An example is shown on the right for the ciphertext "exxegoexsrgi"; the candidate plaintext for shift four "attackatonce" is the only one which makes sense as English text. Another type of brute force attack is to write out the alphabet beneath each letter of the ciphertext, starting at that letter. Again the correct decryption is the one which makes sense as English text. This technique is sometimes known as "completing the plain component".

Another approach is to match up the frequency distribution of the letters. By graphing the frequencies of letters in the ciphertext, and by knowing the expected distribution of those letters in the original language of the plaintext, a human can easily spot the value of the shift by looking at the displacement of particular features of the graph. This is known as frequency analysis. For example, in the English language the plaintext frequencies of the letters E, T, (usually most frequent), and Q, Z (typically least frequent) are particularly distinctive. Computers can automate this process by assessing the similarity between the observed frequency distribution and the expected distribution. This can be achieved, for instance, through the utilization of the chi-squared statistic or by minimizing the sum of squared errors between the observed and known language distributions.

The unicity distance for the Caesar cipher is about 2, meaning that on average at least two characters of ciphertext are required to determine the key. In rare cases more text may be needed. For example, the words "river" and "arena" can be converted to each other with a Caesar shift, which means they can produce the same ciphertext with different shifts. However, in practice the key can almost certainly be found with at least 6 characters of ciphertext.

With the Caesar cipher, encrypting a text multiple times provides no additional security. This is because two encryptions of, say, shift A and shift B, will be equivalent to a single encryption with shift A + B. In mathematical terms, the set of encryption operations under each possible key forms a group under composition.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Caesar cipher


text:
367

Year 367 (CCCLXVII) was a common year starting on Monday of the Julian calendar. At the time, it was known as the Year of the Consulship of Lupicinus and Iovanus (or, less frequently, year 1120 Ab urbe condita). The denomination 367 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * Battle of Solicinium: Emperor Valentinian I launches a punitive expedition against the Alamanni, due to the crises in Britannia and Gaul. The Alamanni re-cross the Rhine and plunder Moguntiacum (modern Mainz).
 * Valentinian I declares that Christians will not be forced into gladiator training schools.
 * Great Conspiracy: The Roman garrison on Hadrian's Wall revolts and allows Picts from Caledonia to devastate Britain. Simultaneously Attacotti, the Scotti from Hibernia (Ireland), and the Saxons from Germania invade the island's mid-western and south-eastern borders. They sack the cities and murder, rape or enslave Romano-British civilians.
 * Eunomius of Cyzicus is banished to Mauretania for harbouring the usurper Procopius.
 * August 4 – Gratian receives the title of Augustus under his father, Valentinian I.
 * Winter – Valentinian I mobilises a massive army for his campaign against the Alamanni and the Franks. He summons the Italian and Illyrian legions for a spring offensive.

Asia

 * The first Korean envoy arrives in Japan, emissary of the government of Kudara.

Religion

 * The first Listing of the New Testament (Bible) is made by St. Athanasius of Alexandria.
 * November 16 – Antipope Ursicinus is banished by the praefecti to Gaul.
 * Epiphanius of Salamis becomes bishop of Salamis, Cyprus.
 * Emperor Valens is baptized by Eudoxius of Antioch.

Science

 * In the region of the constellation Perseus, a star not visible to the naked eye, and 1,533 light years distant from Earth, explodes in a nova. The light from the star, now called GK Persei, was first detected on Earth on February 21, 1901.

Deaths

 * January 13 – Hilary of Poitiers, Byzantine bishop, Doctor of the Church and saint (b. c. 315)

Date unknown

 * Murong Ke, Chinese general and statesman of Former Yan
 * Yang Wu (or Shiqiu), Chinese official, general and regent

title: 367


text:
368

Year 368 (CCCLXVIII) was a leap year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Augustus and Valens (or, less frequently, year 1121 Ab urbe condita). The denomination 368 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * Spring – Emperor Valentinian I and his 8-year-old son, Gratian, cross the Rhine with an army into Alamannic territory. He defeats the Alemanni and burns food stores along the border. A temporary peace is signed with Macrian, king of the Bucinobantes, and Valentinian returns to his capital Augusta Treverorum (modern Trier).
 * Great Conspiracy: Picts, Scotti and Saxons reach Roman London and plunder the city. Theodosius, a general (Comes Britanniarum), is sent with a relief force to Britannia. He marches from Richborough, Kent, to deal with the invaders.
 * Winter – The barbarians are driven back to their homelands, Hadrian's Wall is retaken and order returns to the Roman diocese. Theodosius reorganises the abandoned forts and mounts punitive expeditions in Hibernia (Ireland).

Asia

 * An earthquake strikes Nicaea (modern Turkey).

Births

 * Eustochium, Christian Desert Mother and saint (approximate date)
 * Juqu Mengxun, Chinese prince of the Northern Liang (d. 433)
 * Philostorgius, Anomoean church historian and writer (d. 439)

Deaths

 * July 7 – Maternien, Christian bishop and saint

Date unknown

 * Caesarius of Nazianzus, Roman physician and politician
 * Vulcacius Rufinus, Roman prefect, consul and politician
 * Yue Wan, Chinese general and politician of Former Yan

title: 368


text:
369

Year 369 (CCCLXIX) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Galates and Victor (or, less frequently, year 1122 Ab urbe condita). The denomination 369 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * Spring – Emperor Valens crosses the Danube, and attacks the Gothic tribes (Greuthungi and Tervingi). Valens and Athanaric, the Gothic king, eventually sign a treaty.
 * Fritigern becomes king of the Visigoths; amidst hostilities with his rival Athanaric, he asks Valens and the Thracian field army to intervene. They end the civil war, and Fritigern converts to Christianity.
 * Count Theodosius brings Britain fully back to the Empire after the Great Conspiracy of 367.

Persia

 * King Shapur II occupies the pro-Roman kingdom of Armenia. He besieges Artogerassa in modern Georgia, where Papas (Pap), son of King Arsaces II (Arshak II), defends the fortress and the royal treasure against Persian forces.

Asia

 * Chinese troops of the Jin Dynasty are defeated by Former Yan of the Xianbei.
 * Goguryeo invades Baekje (approximate date).

Art and Science

 * Wulfila creates a Gothic alphabet composed of letters based on Greek and Roman letters, as well as some Germanic runes. He converts the Goths to Arian Christianity.

Births

 * Huan Xuan, Chinese warlord and emperor of the Jin Dynasty (d. 404)

Deaths

 * May 3 – Juvenal of Narni, Roman Catholic bishop, confessor and saint
 * Caesarius of Nazianzus, Byzantine physician and politician (b. 331)
 * Pharantzem, Armenian queen and regent (approximate date)
 * Valentinus, Roman criminal and rebel leader

title: 369


text:
AD 365

Year 365 (CCCLXV) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known in the West as the Year of the Consulship of Augustus and Valens (or, less frequently, year 1118 Ab urbe condita). The denomination 365 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * July 21 – An earthquake and tsunami devastate Crete and Alexandria and affects Italy, Greece, and Palestine.
 * September 28 – Procopius revolts and bribes two legions passing by Constantinople. He proclaims himself Emperor, and takes control of Thrace and Bithynia.
 * November 1 – The Alamanni cross the Rhine and invade Gaul. Emperor Valentinian I moves to Paris to command the army and defend the Gallic cities.

China

 * March 30 – Sixteen Kingdoms: Jin Feidi, age 23, succeeds his brother Jin Aidi as emperor of the Eastern Jin Dynasty. He has no actual power; governmental matters are largely in the hands of his granduncle Sima Yu.

Religion

 * Basil of Caesarea becomes presbyter of Caesarea.
 * Emperor Valens orders the expulsion of the Alexandrian bishop Athanasius from his see, but instead of going into exile Athanasius, now about 67, moves to the outskirts of Alexandria.
 * Antipope Felix II dies after a 9-year reign, ending the double occupancy of the papacy.

Births

 * Julius Agricola, Roman consul and praetorian prefect
 * Kou Qianzhi, Chinese high official and taoist (d. 448)
 * Tao Yuanming, Chinese poet and politician (d. 427)
 * Tufa Rutan, Chinese prince of the Southern Liang (d. 415)

Deaths

 * March 30 – Ai of Jin (or Qianling), Chinese emperor (b. 341)
 * November 22 – Felix II, antipope of Rome (b. 287)
 * Feng Yi (or Zizhuan), Chinese official and general
 * Sima Xun (or Weichang), Chinese warlord (b. 306)
 * Wang Muzhi, Chinese empress of the Jin Dynasty

title: AD 365


text:
364

Year 364 (CCCLXIV) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Augustus and Varronianus (or, less frequently, year 1117 Ab urbe condita). The denomination 364 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * February 17 – Emperor Jovian dies after a reign of eight months. He is found dead in his tent at Tyana (Asia Minor) en route back to Constantinople, in suspicious circumstances.
 * February 26 – Valentinian I is proclaimed Emperor by officers of the Roman army at Nicaea in Bithynia. He addresses the soldiers (who threaten to riot) in a speech. He founds the Valentinianic dynasty and rules the Western portion of the Roman Empire, from Caledonia (Scotland) to the Rhine frontier, ensuring it a few years of relative security. He settles in Paris and establishes a militia to defend the region.
 * March 28 – Valens, brother of Valentinian I, is appointed co-emperor (Augustus) in the palace of Hebdomon (Turkey). He rules the Eastern portion of the Roman Empire, from the Danube to the Persian border, and begins the first anti-pagan persecutions.
 * Britain is forced to endure fierce barbarian raids.

Religion

 * The Council of Laodicea decides some disciplinary questions of the church and attempts to establish the Biblical canon, but fails.

Science

 * Theon of Alexandria, Greek mathematician, observes a solar eclipse (June 16) and a lunar eclipse (November 25). He gains some renown for his version of Euclid's Elements and his commentaries on Ptolemy's Almagest.

Births

 * Blaesilla, Roman noblewoman (d. 384)
 * Huiguo, Chinese Buddhist nun (d. 433)
 * Sima Daozi, Chinese prince and regent (d. 403)
 * Xu Xianzhi, Chinese official and regent (d. 426)

Deaths

 * February 17 – Jovian, Roman emperor (b. 331)
 * Ge Hong, Chinese scholar and taoist (b. 283)
 * Theophilos the Indian, Christian bishop

title: 364


text:
363

Year 363 (CCCLXIII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Iulianus and Sallustius (or, less frequently, year 1116 Ab urbe condita). The denomination 363 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * March 5 – Emperor Julian departs from Antioch with his army (90,000 men) and heads north towards the Euphrates. En route he creates a diversion and sends a force of 30,000 soldiers under his cousin Procopius to Armenia.
 * April – Julian crosses the Euphrates near Hierapolis, using 50 pontoon ships, and moves eastwards to Carrhae. He destroys Perisapora and overruns Persian forts along the desert frontier (Limes Arabicus).
 * May 29 – Battle of Ctesiphon: Julian reaches the vicinity of the strongly fortified capital Ctesiphon. King Shapur II in charge of a large Persian army adopts a scorched earth policy, leaving the Romans desperately short of supplies.
 * June 16 – The Roman army starts its retreat northward to Corduene (Armenia). Julian marches back up the Tigris and burns his fleet of supply ships. During the withdrawal Julian's forces suffer several attacks from the Persians.
 * June 26 – Battle of Samarra: Julian is mortally wounded in a skirmish and dies from a wound received during the fighting near Samarra (Iraq). Jovian, general of the Guard, succeeds him and is proclaimed Emperor by the troops.
 * Emperor Jovian negotiates a disastrous peace with Persia, surrendering four of the five Roman provinces gained by Caesar Galerius in 298, and the cities Nisibis and Singara (Mesopotamia).

Europe

 * The Huns first appear in Europe and reach the Caspian Sea.

Middle East

 * May 18–19 – Galilee earthquake of 363. Petra, capital of the Nabataeans (in modern-day Syria), is seriously damaged.
 * 363 Arsakawan earthquake. It affected the cities of Arsakawan (modern Doğubayazıt) and Salat (modern Sisian).

Astronomy

 * April 20 – The planet Venus occults the planet Jupiter.

Religion

 * The Council of Laodicea, which deals with constricting the conduct of church members, is held. The major canon approved by this council is Canon 29, which prohibits resting on the Sabbath (Saturday), restricting Christians to honoring the Lord on Sunday.
 * Mar Mattai Monastery is founded on Mount Alfaf.

Births

 * Sulpicius Severus, Christian writer (approximate date)
 * Wu Di (or Liu Yu), Chinese emperor of Liu Song (d. 422)

Deaths

 * June 26 – Julian the Apostate, Roman emperor (b. 331)
 * Aemilia Hilaria, Galo-Roman physician and writer (b. 300)
 * Lucillianus, Roman commander (magister equitum)
 * Zhang Xuanjing, Chinese ruler of Former Liang (b. 350)
 * Zhou (or Cheng), Chinese concubine of Jin Chengdi

title: 363


text:
362

Year 362 (CCCLXII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Mamertinus and Nevitta (or, less frequently, year 1115 Ab urbe condita). The denomination 362 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * July 18 – Emperor Julian arrives at Antioch with an expeditionary force (60,000 men), and stays there for nine months, to launch a campaign against the Persian Empire. He secures the co-operation of King Arsaces of Armenia, who fights a bloody guerrilla war against the Persians.
 * Julian builds a fleet of 50 warships, and more than 1,000 transport boats at Samosata, for his expedition in Persia against King Shapur II the Great.
 * An earthquake strikes Nicaea (Turkey).
 * An earthquake strikes Al-Karak (Jordan).

Religion

 * February 21 – Athanasius returns to Alexandria and convenes a council, at which he appeals for unity among Christians who differ in terminology, but Emperor Julian orders Athanasius to leave Alexandria. He will remain in exile in Upper Egypt, until after Julian's death the next year.
 * October 22 – The temple of Apollo at Daphne, outside Antioch, is destroyed in a mysterious fire.

Births

 * Flavia Maxima Constantia, daughter of Constantius II (approximate date)
 * Mesrop Mashtots, Armenian linguist (d. 440)
 * Jin Xiaowudi, emperor of the Eastern Jin Dynasty (d. 396)

Deaths

 * January 1 – Paulus Catena, Roman politician
 * February 25 – Reginos, Greek Orthodox bishop and saint
 * May 10 – Gordianus and Epimachus, Roman Catholic priests, martyrs and saints
 * June 5 – Dorotheus of Tyre, Roman Catholic bishop and martyr (b. 255)
 * June 27 – Crispus, Crispinianus, and Benedicta, Roman Catholic, priests, martyrs and saints
 * June 28 – Basil of Ancyra, Byzantine Orthodox bishop and saint
 * August 5 – Eusignius of Antioch, Byzantine Orthodox bishop and martyr (b. 252)
 * August 7 – Donatus of Arezzo, Roman Catholic bishop and saint
 * October 16 – Eliphius, Roman Catholic priest and saint
 * October 20
 * Artemius, Egyptian Orthodox martyr and saint
 * Theodoret of Antioch, Syrian Christian priest and martyr
 * December 10 – Gemellus of Ancyra, Byzantine Orthodox bishop and saint

Date unknown

 * Princess Dowager Ma, concubine of Zhang Jun

title: 362


text:
361

Year 361 (CCCLXI) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Taurus and Florentius (or, less frequently, year 1114 Ab urbe condita). The denomination 361 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * November 3 – Emperor Constantius II dies of a fever at Mopsuestia in Cilicia, age 44; on his deathbed he is baptised, and declares his cousin Julian the Apostate rightful successor.
 * December 11 – Julian becomes sole emperor of the Roman Empire; he rules from Constantinople, and tries to restore paganism. Constantius II is buried in the Church of the Holy Apostles.
 * Ministers and followers of Constantius II are put on trial, at the Chalcedon Tribunal.

China

 * July 10 – Sixteen Kingdoms: Jin Aidi, age 20, succeeds Jin Mudi, as emperor of the Eastern Jin Dynasty.

Art

 * 361–363 – A Julian the Apostate coin is issued. It is now kept at the British Museum, London.

Medicine

 * Constantinople enforces a strict licensing system for physicians.

Religion

 * Emperor Julian tries to organize a pagan church and substitute it for Christianity. Pope Liberius repudiates the Arian creed, and declares that the Council of Arminium has no authority to issue decrees.
 * Gregory Nazianzus (Saint Gregory the Theologian) returns to Nazianzus and is appointed a priest by his father, who wants him to assist local Christians.
 * Construction of the Monastery of Saint Anthony in the Eastern Desert of Egypt begins.
 * December 24 – George of Cappadocia, the Arian intruding bishop of Alexandria, is murdered in his see and Athanasius of Alexandria returns to his native city in triumph.

Deaths

 * November – Constantius II, Roman Emperor (b. 317)
 * December 24
 * George of Cappadocia, Byzantine Orthodox archbishop and saint
 * George of Laodicea, Byzantine Orthodox archbishop and saint

Date unknown

 * Apodemius, Roman officer and secret agent
 * Eusebius, Roman officer
 * Jin Mudi, emperor of the Eastern Jin Dynasty (b. 343)
 * Li Shi, emperor of the Chinese Ba-Di state Cheng Han
 * Maximus of Naples, Roman Catholic archbishop and saint
 * Song Hun, regent of the Chinese state Former Liang
 * Wang Xizhi, Chinese calligrapher (b. 303)
 * Zhu Jingjian, Chinese Buddhist nun (b. 292)

title: 361


text:
AD 360

Year 360 (CCCLX) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Constantius and Iulianus (or, less frequently, year 1113 Ab urbe condita). The denomination 360 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years.

Roman Empire

 * February – Julian, Roman Caesar, is proclaimed emperor by the Gallic legions in Lutetia (modern Paris), at the Thermes de Cluny. They refuse to support the eastern campaign against King Shapur II of Persia, and revolt.
 * The Alamanni raid Raetia (Switzerland), but are pushed back behind the Rhine by Julian, into the Black Forest.
 * King Shapur II continues his campaign against the Roman fortresses, capturing Singara, Bezabde and Nisibis.
 * Emperor Constantius II and Julian exchange several letters, both hoping to avoid a civil war.

Europe

 * The Huns invade Europe by the thousands, spreading terror as they take over territories held for generations by Alans, Heruls, Ostrogoths and Visigoths.

Agriculture

 * Roman authorities in Britain export wheat to supply the legions on the Rhine; they have encouraged production of wheat for that purpose.

Religion

 * Council of Constantinople (360): Emperor Constantius II requests a church council, at Constantinople; both the eastern and western bishops attend the meeting. Ulfilas also attends the council and endorses the resulting creed. After the council several homoiousian bishops are deposed or banished, including Macedonius I of Constantinople and Cyril of Jerusalem.
 * At about this date, Ligugé Abbey in France is founded for the monastic Order of Saint Benedict by Martin of Tours, under dispensation from Bishop Hilary of Poitiers.

Births

 * John Cassian, Desert Father and Christian saint (approximate date)
 * Saint Mesrob, Armenian monk and theologian (approximate date)
 * Saint Ninian, missionary to Scotland (approximate date)
 * Tao Sheng, Chinese Buddhist scholar (approximate date)
 * Wang Fahui, empress during the Jin Dynasty (d. 380)

Deaths

 * May 4 – Judas Cyriacus, Roman Catholic priest and saint
 * October 14 – Gaudentius of Rimini, Roman Catholic priest and saint
 * November 11 – Abramios the Recluse, Byzantine Orthodox priest and saint
 * December 18 – Auxentius of Mopsuestia, Byzantine Orthodox priest and saint

Date unknown

 * Eusebius of Emesa, Byzantine Orthodox bishop and saint (b. 290)
 * Murong Jun, emperor of the Former Yan (b. 319)

title: AD 360