C Édric Milliet

FIELDS AND RINGS WITH FEW TYPES

Keywords: 2000 Mathematics Subject Classification. 03C45, 03C60 Small, weakly small, field, ring, Artin-Schreier, Cantor-Bendixson rank, local descending chain condition

published or not. The documents may come

In [START_REF] Milliet | On properties of (weakly) small groups[END_REF], the author has begun the exploration of small and weakly small groups. He noticed that a weakly small group G inherits locally several properties that omega-stable groups share globally. For instance G satisfies local descending chain conditions. Every definable subset of G has a local stabiliser with good local properties. If G is infinite, it also possesses an infinite abelian subgroup, not necessarily definable though. Let's not forget that weakly small structures include omegastable ones but also ℵ 0 -categorical, minimal, and d-minimal ones. The following pages aim at classifying weakly small fields and rings, bearing in mind the classification of the particular cases cited above. The guiding line is that the formers should not differ much from the latters, at least locally, in the following sense :

Definition. Let M be a first order structure and P any property. M is said to be locally P if every finitely generated algebraic closure in M has property P .

Let us bring to mind some known results about omega-stable, ℵ 0 -categorical, and minimal rings. First concerning fields : an ℵ 0 -categorical field is finite. Macintyre showed in 1971 that an omega-stable field is either finite or algebraically closed [START_REF] Macintyre | On ω 1 -categorical theories of fields[END_REF]. Wagner drew the same conclusion for a small field, as well as for a minimal field of positive characteristic [START_REF] Frank | Small fields[END_REF][START_REF] Frank | Minimal fields[END_REF]. Poizat extended the latter to d-minimal fields of positive characteristic [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes de et corps de dimension un et de déterminer leurs propriétés algébriques[END_REF]. Whether the same result holds even for a minimal field of characteristic zero is still unknown. We begin Section 2 by giving another proof that a small field is either finite or algebraically closed, and derive that for an infinite weakly small field F , no field extension of F of finite degree has Artin-Schreier extensions. It follows that a weakly small field of characteristic 2 is finite or algebraically closed. Wagner had already noticed in [START_REF] Frank | Small fields[END_REF] that a weakly small field is either finite or has no Artin-Schreier nor Kummer extensions. As a field extension of finite degree of a weakly small field has no obvious reason to be weakly small, our statement is a non-trivial improvement of [START_REF] Frank | Small fields[END_REF]. In section 3, we do not assume commutativity anymore and show that a weakly small division ring of positive characteristic is locally finite dimensional over its centre. Recall that superstable division rings [START_REF] Cherlin | Superstable fields and groups[END_REF]Cherlin,Shelah] and even supersimple ones [START_REF] Pillay | Supersimple fields and division rings[END_REF]Pillay,Scanlon,Wagner] are known to be fields.

We then turn to small difference fields. Hrushowski proved that in a superstable field, any definable field morphism is either trivial or has a finite set of fixed points [START_REF] Hrushovski | On Superstable Fields with Automorphisms, The model theory of groups[END_REF]. We show that this also holds for a small field of positive characteristic. We finish by rings in section 5. The structure theory of an associative ring R breaks usually into three parts : the study of its Jacobson radical ring J ; the study of the quotient ring R/J, the so-called reduced ring ; and the gluing back process, buiding a ring with a given radical and reduced rings in the spirit of Wedderburn-Malcev's theorem []. An omega-stable ring R is known to have a nilpotent Jacobson radical J and R/J is a finite product of matrix rings over finite or algebraically closed fields [START_REF] Cherlin | Categoricity and stability of commutative rings[END_REF][START_REF] Macintyre | On ω 1 -categorical theories of fields[END_REF]Cherlin,Reineke,Macintyre]. The Jacobson radical of an ℵ 0 -categorical ring is nilpotent [START_REF] Cherlin | On ℵ 0 -categorical nilrings[END_REF][START_REF] Cherlin | On ℵ 0 -categorical nilrings II[END_REF]Cherlin]. For a weakly small ring R, we show that its radical is locally nilpotent (hence nil). Moreover, R is locally modulo its radical the product of finitely many matrix rings over division rings.

(Weakly) small tools

Let us recall what a small theory and more generally a weakly small structure are. Definition 1.1. A theory is small if it has countably many complete n-types without parameters (or equivalently over any fixed finite parameter set) for every natural number n. A structure is small if so is its theory.

Definition 1.2 (Belegradek).

A structure is weakly small if for any of its subsets A, there are countably many complete 1-types over A.

For convenience of the reader, we state here the main results of [START_REF] Milliet | On properties of (weakly) small groups[END_REF] that will be needed in the sequel. We refer to the latter paper the reader willing to know more about weakly small groups.

In a weakly small structure M , for any finite parameter subset A of M , the space S 1 (A) of complete 1-types over A is a countable compact Hausdorff space. It has an ordinal Cantor-Bendixson rank and one can compute the Cantor-Bendixson rank over A of any of its element p. We write it CB A (p). For any A-definable set X of arity 1, we write CB A (X) for the maximum Cantor rank of the complete 1-types over A countaining the formula defining X. Only a finite number of complete 1types over A with same CB A -rank as X do contain the formula defining X. We call this natural number the Cantor-Bendixson degree of X over A, and write it dCB A (X). What has been said for 1-types of a weakly small structure is also valid for every n-type of a small structure over parameters in an arbitrary finite set. The following two Lemmas hold in any structure.

Lemma 1.3. Let X and Y be A-definable sets. Let f be an A-definable map from X onto Y . If the fibres of f have no more than n elements, then f preserves the Cantor rank over A. Moreover,

dCB A (X) ≤ dCB A (Y ) ≤ n • dCB A (X)
Lemma 1.4. Let X be a ∅-definable set, and a an element algebraic over the empty set. Then CB a (X) equals CB ∅ (X). Lemma 1.4 allows to define the local Cantor rank of an a-definable set X, to be its Cantor rank over any b defining X and having the same algebraic closure as a. We shall write acl(B) for the algebraic closure of some set B, and dcl(B) for its definable closure.

Theorem 1.5 (Weakly small descending chain condition). In a weakly small group, the trace over acl(∅) of a descending chain of acl(∅)-definable subgroups becomes stationary after finitely many steps.

For any ∅-definable set X in a weakly small group G, if Γ is the algebraic closure of a finite tuple g from G, one can define the local almost stabiliser of X in Γ to be

Stab Γ (X) = {x ∈ Γ : CB x,g (xX∆X) < CB g (X)} Stab Γ (X) is a subgroup of Γ. If δ is any subgroup of Γ, we write Stab δ (X) for Stab Γ ∩ δ.
Here is a local analogue of what happens for the stabiliser of a definable set of maximal Morley rank in an omega-stable structure : Proposition 1.6. Let G be a weakly small group, g a finite tuple of G, and X a g-definable subset of X. If δ is a subgroup of dcl(g) and if X has maximal Cantor rank over g, then Stab δ (X) has finite index in δ.

Next proposition can be found in [START_REF] Milliet | Small Skew fields[END_REF]. Proposition 1.7. Let G be a small group, and f a definable group homomorphism of G. There exists a natural number n such that Kerf n • Imf n equals G.

Weakly small fields

We begin by proposing a lightened version of a result from Wagner. Lemma 2.1. An infinite weakly small field is perfect.

Proof. Let f be a group homomorphism of some weakly small group G. Suppose that f has finite kernel of cardinal n. An easy consequence of Lemma 1.3 is that the image of f has index at most n in G. It follows that the image of the Froboenius map is a sub-field having finite additive (and multiplicative !) index. Theorem 2.2. A weakly small infinite field, possibly skew, has no definable additive nor multiplicative subgroup of finite index.

Proof. Let K be this field and let H be a definable additive subgroup of K having finite index. Suppose first that there is an infinite finitely generated algebraic closure Γ. Note that Γ is a field. The intersection of λH ∩ Γ where λ runs over Γ is a finite intersection by Theorem 1.5 hence has finite index in Γ. It is also an ideal of Γ and must equal Γ. So Γ is included in H. As this holds for any infinite finitely generated Γ, the group H equals K.

Otherwise, K is locally finite. By Wedderburn's theorem, K is commutative and equals acl(∅). According to Theorem 1.5, it satisfies the descending chain condition on definable subgroups. K has a smallest definable additive subgroup of finite index, which must be an ideal, and hence equals K.

For the multiplicative case now, let us consider a multiplicative subgroup M of K × having index n. Let us suppose first that there is δ an infinite finitely generated sub-field of K. There is no harm in extending δ so that each coset of M be δdefinable. M has maximal Cantor rank over δ by Lemma 1.3, so its almost additive stabiliser in δ has finite additive index in δ by Proposition 1.6, as well as the almost stabiliser of any of its cosets. So the almost stabiliser of all the cosets is an ideal of δ having finite index, hence equals the whole of δ. We finish as Poizat in [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes de et corps de dimension un et de déterminer leurs propriétés algébriques[END_REF] : we have just shown that 1 + aM ≃ aM for every coset aM , where ≃ stands for equality up to small Cantor rank over δ. For every coset aM , and every x in aM but a small ranked set, 1 + x belongs to aM , so x -1 + 1 ∈ M , and the complement of M has a small Cantor rank : M is exactly K × . Otherwise K is locally finite and has characteristic p. By Lemma 2.1, the group K × is p-divisible, so K × can not have a proper subgroup of finite index.

Before going further, let us remind the reader with a few definitions. Let L/K be a field extension. It is a Kummer extension if it is generated by K and one nth root of some element in K. It is an Artin-Schreier extension, if L is generated by K and one x such that x px belongs to K (x is called a pseudo-root of K). Fact 2.3 (Artin-Schreier, Kummer [START_REF] Lang | Algebra[END_REF]). Let K be a field of characteristic p (possibly zero) and L a cyclic Galois extension of finite degree n.

(i) Suppose that p is zero, or coprime with n. If K contains n distinct nth roots of 1, then L/K is a Kummer extension. (ii) If p equals n, then L/K is an Artin-Schreier extension.

Corollary 2.4 (Wagner [START_REF] Frank | Small fields[END_REF]). A small field is finite or algebraically closed.

Proof. Let K be a small infinite field. For every natural number n, the nth power map has bounded fibres so its image has finite index by Lemma 1.3, and the map is onto by Theorem 2.2. So is the map mapping x to x p -x. We conclude as Macintyre in [START_REF] Macintyre | On ω 1 -categorical theories of fields[END_REF] for omega-stable fields : first of all, K contains every root of unity. For if a is a nth root of 1 not in K with minimal order, K(a)/K has degree m < n. By minimality of n, every mth root of unity is in K. If m is zero or coprime with the characteristic of K, then K(a) is of the form K(b) with b m ∈ K after Fact 2.3. As K × is divisible, b is in K, a contradiction. So K has positive characteristic p, and p divides m. K(a) contains an extension of degree p over K, which is an Artin-Schreier extension after Fact 2.3, a contradiction.

Secondly, if K is not algebraically closed, it has a normal extension L of finite degree n, which must be separable as K is perfect (Lemma 2.1) ; its Galois group contains a cyclic sub-group of prime order q, the invariant field of which we call M . Note that L is interpretable in a finite Cartesian power of K, so is small too. If q is not the characteristic, as K contains every qth root of 1, the extension L/M is Kummer ; if q equals the characteristic, L/M is an Artin-Schreier extension, a contradiction in both cases.

Note that the first part of the previous proof still holds for weakly small fields : Corollary 2.5. An infinite weakly small field contains every root of unity, hence has no solvable by radical extension.

Proof. If the extension L/K is solvable by radical, there is a tower of fields

K = K 0 ⊂ K 1 ⊂ • • • ⊂ K n = L
so that each K i+1 /K i be generated by either a nth root or a pseudo-root of some element K i . But K 0 has no Artin-Schreier or Kummer extension by Corollary 2.4.

Note however that, as an algebraic extension of a weakly small field has no obvious reason to be weakly small, we cannot apply Macintyre's argument to deduce that a weakly small field is finite or algebraically closed. Nevertheless, stepping on the additive structure of the field, we can show that every algebraic extension of an infinite weakly small field is Artin-Schreier closed. This is a first step towards Problem 12.5 in [START_REF] Frank | Small fields[END_REF] asking whether an infinite weakly small field is algebraically closed.

Definition 2.6. An Abelian structure is any abelian group together with predicates interpreting subgroups of its finite Cartesian powers.

As for a pure module, an Abelian structure has quantifier elimination up to positive prime formulas (see [START_REF] Volker | Quantifier elimination for abelian structures[END_REF]Weispfenning], or [24, Theorem 4.2.8]) :

Fact 2.7 (Weispfenning). In an Abelian structure A, a definable set is a boolean combination of cosets of acl(∅)-definable subgroups of Cartesian powers of A.

An Abelian structure A is stable : let us take a formula ϕ(x, y) such that ϕ(x, 0) defines an acl(∅)-definable subgroup of A n , and ϕ(x, y) the coset of y. Two formulas ϕ(x, a) and ϕ(x, b) define cosets of the same group, so they must be equal or disjoint. It follows that there are countably many ϕ-types over any countable set of parameters. By Fact 2.7, this is sufficient to show that A is stable. In a stable structure, we call a dense forking chain any chain of complete types p q indexed by Q such that for every rational numbers q < r, the type p r be a forking extension of p q .

Stable theories with no dense forking chains have been introduced in [START_REF] Pillay | Countable models of 1-based theories[END_REF]Pillay]. They generalise superstable ones. In a stable structure M with no dense forking chains, every complete type (and not only 1-types !) has an ordinal dimension, and for any dimension α, a Lascar α-rank. We shall write dim(p) for the dimension of the type p, and U α (p) for its α-rank. They are defined as follows : Definition 2.8 (Pillay). For two complete types p ⊂ q, let us define the dimension of p over q written dim(p/q) by the following induction.

• dim(p/q) is -1 if q is a nonforking extension of p.

• dim(p/q) is at least α + 1 if there are non forking extensions p ′ and q ′ of p and q, and infinitely many complete types p 1 , p 2 , . . . such that p ′ ⊂ p 1 ⊂ p 2 ⊂ • • • ⊂ q ′ and dim(p i /p i+1 ) ≥ α for all natural number i. • dim(p/q) is at least λ for a limit ordinal λ if dim(p/q) ≥ α for all α < λ. Definition 2.9 (Pillay). For a complete type p, we set dim(p) = dim(p/q) where q is any algebraic extension of p. Definition 2.10 (Pillay). For every ordinal α, we define inductively the U α -rank of a complete type p by

• U α (p) is at least 0. • U α (p) is at least β + 1 if there is an extension q of p such that dim(p/q) ≥ α and U α (q) ≥ β. • U α (p) is at least λ for a limit ordinal λ if U α (p) ≥ β for all β < λ.
To any type-definable stable group in M can be associated the U α -rank and the dimension of any of its generic types over M . We refer the reader to [17, 8, Herwig, Loveys, Pillay, Tanović, Wagner] for more details. We shall just recall two facts : the Lascar inequalities which are still valid for the U α -rank, as well as their group version ; and the link between the U α -rank and the existence of a dense forking chain.

Fact 2.11 (Lascar inequalities for U α -rank [START_REF] Pillay | Countable models of 1-based theories[END_REF][START_REF] Herwig | Stable theories without dense forking chains[END_REF]).

(1) In a stable structure, for every tuple a, b and every set A,

U α (b/Aa) + U α (a/A) ≤ U α (ab/A) ≤ U α (b/Aa) ⊕ U α (a/A)
(2) For any type-definable group G in a stable structure, and any type-definable subgroup H of G,

U α (H) + U α (G/H) ≤ U α (G) ≤ U α (H) ⊕ U α (G/H)
Proof. We only prove point (2), which does not appear anywhere to the author's knowledge but follows from [START_REF] Cherlin | On ℵ 0 -categorical nilrings[END_REF]. Note that passing from the ambient structure M to M heq , one can use hyperimaginary parameters in [START_REF] Cherlin | On ℵ 0 -categorical nilrings[END_REF].

Let tp(a/M ) be a generic type of G. We write a H the hyperimaginary element which is the image of a in G/H. In a κ-saturated stable structure M , for any formula ϕ(x, y), we can compute the Cantor-Bendixson of the topological space S ϕ (M ) whose elements are the complete ϕ-types over M . Let ψ(x) be another formula and S ϕ,ψ the subset of S ϕ (M ) whose elements are consistent with ψ. It is a closed subset of S ϕ (M ). The local ϕ-rank of ψ is the Cantor-Bendixson rank of S ϕ,ψ . We write it CB ϕ (ψ). The local ϕ-rank of a type p is the minimum local ϕ-rank of the formulas implied by p. If M is a stable group, the stratified ϕ-rank of ψ is its φ-rank, where φ(x, y) stands for the formula ϕ(y 2 • x, y 1 ). We write it CB * ϕ (ψ). In a κ-saturated stable group G, let H and L be two type-definable subgroups. H and L are commensurable if the index of their intersection is bounded (i.e. less that κ) in both of them. Recall that this is equivalent to H and L having the same stratified ϕ-rank for every formula ϕ. Theorem 2.13. Let be an Abelian structure with weakly small universe. Its theory has no dense forking chain. Remark 2.14. Pillay showed that a small 1-based structure has no dense forking chain [START_REF] Pillay | Countable models of 1-based theories[END_REF]Lemma 2.1]. In particular, a small Abelian structure has no dense forking chain either. The difficulty of Theorem 2.13 comes from the fact that weak smallness does not bound a priori the number of pure n-types for n ≥ 2, which is a crucial assumption in the proof of [START_REF] Pillay | Countable models of 1-based theories[END_REF]Lemma 2.1].

Proof. According to Fact 2.12, one just needs to show that for all finite tuple a and all set A, there is an ordinal α such that U α (a/A) is ordinal. Note that the first of Lascar inequalities for the U α -rank implies that U α ((a 1 , . . . , a n )/A) is less or equal to U α ((a 2 , . . . , a n )/Aa 1 ) ⊕ U α (a 1 /A). So, by induction on the arity of a, and Fact 2.12 again, we may consider only 1-types, and suppose for a contradiction that there be a dense forking chain of arity 1.

(1) We first claim that there exists a dense ordered chain (H i ) i∈Q of acl(∅)-typedefinable pairwise non commensurable subgroups.

Let (tp(a/A i )) i∈Q be a dense forking chain, that is A i is included in A j and tp(a/A j ) forks over A i for all i < j. By Fact 2.7 , every formula appearing in tp(a/A i ) is a boolean combination of cosets of acl(∅)-definable groups. There is a smallest acl(∅)-type-definable group H i such that the type tp(a/A i ) contains the formulas defining aH i . If CB * ϕ (a/A i ) < CB * ϕ (aH i ) for some formula ϕ, then there is an acl(∅)-definable subgroup G i with the formula defining aG i included in tp(a/A i ), and

CB * ϕ (aG i ) < CB * ϕ (aH i ). This implies CB * ϕ (G i ) < CB * ϕ (H i )
and contradicts the minimality of H i . It follows that tp(a/A i ) is a generic type of aH i . Moreover, aH i is A i -type-definable. For all i < j, the type tp(a/A j ) forks over A i so there must be a formula ϕ such that aH j and aH i have different stratified ϕ-ranks. Then, one has CB * ϕ (H i ) < CB * ϕ (H j ) so H i and H j are non-commensurable groups.

(2) Let us now build 2 ℵ0 complete 1-types over ∅.

As the structure is stable, each H i is the intersection of acl(∅)-definable groups H ij . Let H ij stand for the ∅-definable union of the conjugates of H ij under Aut(∅). Let us call H i the ∅-type-definable intersection of the H ij over j. For every real number r, we call p ′ r the partial type defining i≥r H i and p r the following partial type

p ′ r ∪ {ψ : ψ formula over ∅ with CB * ϕ (¬ψ) < CB * ϕ (p ′ r )
for some ϕ} Note that every formula ψ in the second part of the type p r above is contained in every generic type of the structure (and also in the generic types of i≥r H i ). Every p r is thus consistent. We claim that if r = q, then p r and p q are inconsistent. In any stable group, if G 1 , G 2 , . . . are decreasing definable subgroups, for every formula ϕ, there is an index i ϕ such that the equality CB * ϕ ( i≥1 G i ) = CB * ϕ (G j ) holds for all j > i ϕ . So one can find two indexes i and j (depending on ϕ) such that all the following equalities hold

CB * ϕ ( i≥r H i ) = CB * ϕ ( H i ) = CB * ϕ (H ij ) CB * ϕ ( i≥r H i ) = CB * ϕ (H i ) = CB * ϕ (H ij )
As the stratified ϕ-ranks are preserved under automorphisms, and as the rank of a finite union equals the maximum of the ranks, we get

CB * ϕ ( i≥r H i ) = CB * ϕ ( i≥r H i )
By point (1), the groups H i are pairwise non-commensurable. It follows from the last equality that every pair of elements of the chain ( H i ) i∈Q has at least one CB * ϕrank distinguishing them. So the types (p r ) r∈R are pairwise inconsistent. We may complete them and build 2 ℵ0 complete 1-types.

Note that if G is a stable group with no dense forking chain, and f a definable group morphism from G to G with finite kernel, then G and f (G) have same dimension and U α -rank. This follows from the second Lascar inequality applied to G and KerG. More generally : Lemma 2.15. Let X and Y be type-definable sets, and let f be a definable map from X onto Y the fibres of which have no more than n elements for some natural number n. Then X and Y have the same dimension, and same U α -rank for every ordinal α.

Proof. One just needs to notice that if q is some type in X, and if p is an extension of q, then p is a forking extension of q if and only if f (p) is a forking extension of f (q). Proposition 2.16. Let G be a stable group without dense forking chain, and let f be a definable group morphism from G to G. If f has finite kernel, its image has finite index in G.

Proof. Let us write H for f (G), and let us apply the first Lascar equality. We get U α (H) + U α (G/H) ≤ U α (G). But H and G have the same U α -rank after Lemma 2.15. It follows that U α (G/H) is zero. This holds for every ordinal α, so dim(G/H) is -1. This means that G/H is finite.

Remark 2.17. In Proposition 2.16, one cannot bound the index of the image of f with the cardinal of its kernel. Consider for instance the superstable group (Z, +), and the maps f n mapping x to the n times sum x + • • • + x, when n ranges among natural numbers.

Corollary 2.18. Every algebraic extension of an infinite weakly small field is Artin-Schreier closed.

Proof. Let K be an infinite weakly small field of positive characteristic p, let L be an algebraic extension of K and f the Artin-Schreier map from L to L. We consider the additive structure of L, together with f : it is an Abelian structure with weakly small universe K. It has no dense forking chain by Theorem 2.13. The map f has finite fibres so f (L) has finite additive index in L by Proposition 2.16. But K has no proper definable additive subgroup of finite index by Theorem 2.2, so neither has any finite Cartesian power of K, thus f is onto.

Corollary 2.19. The degree of an algebraic extension of an infinite weakly small field of positive characteristic p is not divisible by p.

Proof. Let K be this infinite weakly small field ; it is perfect by Theorem 2.2. If there is an algebraic extension the degree of which is divisible by p, there is also a normal separable extension L of finite degree divisible by p. Its Galois group has a subgroup of order p, the invariant field of which we note K 1 . The extension L/K 1 is an Artin-Schreier extension, a contradiction. Corollary 2.20. A weakly small field of characteristic two is either finite or algebraically closed.

Proof. If it is infinite and not algebraically closed, it has a normal separable algebraic extension of finite degree. According to Corollaries 2.19 and 2.5, its Galois group neither has even order, nor is soluble, a contradiction to Feit-Thomson's Theorem.

Weakly small division rings

Recall that a superstable division ring is a field [START_REF] Cherlin | Superstable fields and groups[END_REF]Cherlin,Shelah]. The author has shown that a small division ring of positive characteristic is a field [START_REF] Milliet | Small Skew fields[END_REF]. It is still unknown whether this extends to weakly small division rings. In this section we show, at least, that every finitely generated algebraic closure in a weakly small division ring has finite dimension over its centre. With the previous section, this implies that a weakly small division ring of characteristic 2 is a field. From now on, let D be an infinite weakly small division ring. If K is a definable subdivision ring of D, one may view D as a left or right vector space over K. However, we will not distinguish between the left and right K-dimension of D thanks to : Lemma 3.1. If K is a definable sub-division ring of D, and if D has finite left or right dimension over K, then D has finite right and left dimension over K, and those dimensions are the same. Moreover, there is a set which is both a left and right K-basis of D.

Proof. Let f 1 , . . . , f n be a left and right K-free family from D, with n maximal. Let F r and F l be the set of respectively right and left linear K-combinations of the f i . If F r < D and F l < D, then F r ∪F l < D, a contradiction with n being maximal. So suppose that D equal F r . The group homomorphism from D + mapping a right decomposition n i=1 f i k i to n i=1 k i f i is a definable embedding, hence surjective after Lemma 1.3. Thus F l , F r and D are equal. Proposition 3.2. The centre of an infinite weakly small division ring is infinite.

Proof. We may assume that D has non-zero characteristic, as this obviously holds in zero characteristic. We may also assume that D is not locally finite and has an element b of infinite order. It follows from Corollary 2.5 that Z(C(b)) contains every root of 1. We claim that all those roots are in Z(D). Suppose not, and let a be non central with a q = 1 and q a prime number. According to a lemma of Herstein [7, Lemma 3.1.1], there exists a natural number n and an x in D with xax -1 = a n but a n = a . If x has finite order, the division ring generated by x and a is finite, a contradiction to Wedderburn's Theorem. So x has infinite order. Conjugating q -1 times by x, we get x q-1 ax -q+1 = a n q-1 = a. Note that x q-1 has infinite order, so Z(C(x q-1 )) contains x by Corollary 2.5. It follows that a and x commute, a contradiction. Corollary 3.3. An element and a power of it have the same centraliser.

Proof. Let a be in D. We obviously have C(a) ≤ C(a n ). Conversely, by Proposition 3.2, the field Z(C(a n )) is infinite. Corollary 2.5 implies that it contains a. Remark 3.4. It follows that every element having finite order lies in the centre. Similarly, for every non constant polynomial P with coefficients in the centre having a soluble Galois group, P (a) and a have the same centraliser in D. If D is in addition small, this holds for every non constant polynomial with coefficients in the centre. x commutes with a and let y be in acl(a). For every natural number m, the elements y x m and y are conjugated by the action of the automorphisms group fixing a pointwise. So there must be two distinct natural numbers n and m so that y x n and y x m be equal : y commutes with a power of x, hence with x by Corollary 3.3. Lemma 3.6. Let γ stand for the conjugation map by some a in D. For all λ in D, the kernel of γλ.id is a C(a)-vector space having dimension at most 1.

Proof. Let some non zero x and y be in the kernel of γλ.id. The equalities x a = λx and y a = λy yield (y -1 x) a = y -1 x. Lemma 3.7. In a weakly small division ring of positive characteristic, for all a, every finitely generated algebraic closure Γ containing a is a finite dimensional C Γ (a)-vector space.

Proof. We write f for the endomorphism mapping x to x ax. Let K and H stand for the kernel and the image of f respectively. Note that f is not onto, as otherwise there would be some x verifying x a = x + 1 and x a p = x + p = x, a contradiction to Lemma 3.3. Let f be the restriction of f from D + /K to D + /K. The set H is a K-vector space so the intersection H ∩ K is an ideal of K, which must be trivial f is not onto. The map f is injective hence surjective, so we get

D = H ⊕ K. This yields Γ = H ∩ Γ ⊕ K ∩ Γ
The intersection I of the sets λH ∩ Γ where λ runs over Γ is a finite intersection, of size n say : it is a left ideal of Γ, hence zero. But H ∩ Γ is a K ∩ Γ-vector space having codimension 1, so I has codimension at most n.

Theorem 3.8. A weakly small division ring of positive characteristic is locally finite dimensional over its centre.

Proof. Let Γ be finitely generated algebraic closure, and D 0 , . . . D n+1 a maximal chain of centralisers of elements in Γ such that the chain Corollary 3.9. A small division ring of positive characteristic is a field.

Γ > D 1 ∩ Γ > • • • > D n ∩ Γ > D n+1 ∩ Γ
Proof. Let Γ be the algebraic closure of a finite tuple a. By Corollary 3.5, we have

Z(Γ) = Z(C(Γ)) ∩ Γ = Z(C(a)) ∩ Γ
By [START_REF] Frank | Small fields[END_REF], Z(C(a)) is algebraically closed. It follows that Z(Γ) is relatively algebraically closed in Γ, so a small division ring is locally commutative, hence commutative.

Corollary 3.10. A weakly small division ring of characteristic 2 is a field.

Proof. Follows from Corollary 2.20 with the same proof as Corollary 3.9.

Corollary 3.11. Vaught's conjecture holds for the pure theory of a positive characteristic division ring.

Proof. If the theory of an infinite pure division ring has fewer than 2 ℵ0 denumerable models, it is small : it is the theory of a algebraically closed field, which has countably many denumerable models as noticed in [START_REF] Frank | Small fields[END_REF].

In positive characteristic, we can just say the following : Proposition 3.12. If D is small, let a be outside the centre, and write γ for the conjugation by a. For all non-zero polynomial a n X n +• • •+a 1 X +a 0 with coefficients in the centre of D, the morphism a n γ n + • • • + a 1 γ + a 0 Id is onto.

Proof. Let K be the field C(a). As Z(D) is algebraically closed, P splits over Z(D).

As a product of surjective morphisms is still surjective, it suffices to show the result for some irreducible P . Let λ be in the centre, let f be the morphism γλ.id, and let t be outside the image of f . The map f is a K-linear map ; its kernel must be a line or a point. According to Proposition 1.7, we get D = Kerf m + Imf m for some natural number n. Set H the image of f m , and L its kernel. Note that L has finite K-dimension. We may replace L by a definable summand of H, and assume that L and H be disjoints. Let Γ an infinite finitely generated algebraic closure containing t, a, some b which does not commute with a, and the K-basis of L. We still have

Γ = L ∩ Γ ⊕ H ∩ Γ
The intersection I of the sets λH ∩ Γ, where λ runs over Γ is a finite intersection by Theorem 1.5 : it is an ideal of Γ which does not contain t, hence zero. But H ∩ Γ has finite K ∩ Γ-codimension, hence so has I. According to [5, Corollary 2 p.49],

we have [Γ :

K ∩ Γ] = [Γ : C Γ (a)] = [Z(Γ)(a) : Z(Γ)] < ∞ But Z(Γ) is nothing more than Z(C D (Γ)) ∩ Γ. By Corollary 3.5, Z(C D (Γ)
) is an algebraically closed field so a belongs to Z(Γ), a contradiction.

Corollary 3.13. In a small field, the conjugation by any element generates a central division algebra.

Small difference fields

It is a common phenomenon to weakly small and superstable groups, and to stable groups without dense forking chain, that when a definable group homomorphism has a somehow small kernel, its image has to be somehow big. See [20, Proposition 1.7], or [START_REF] Frank | Small stable groups and generics[END_REF]Corollary 6]. As for a definable endomorphism of a small field, we have the following :

Proposition 4.1. Let K be a small infinite field, F a definable subfield, and f a non trivial F -linear endomorphism of K, the kernel of which has finite F -dimension. Then f is onto.

Proof. By Proposition 1.7, the equality K = Kerf m + Imf m holds for some natural number m. Let H be the image of f m and L its kernel. Note that if n is the dimension of Kerf over F , then Kerf m has dimension at most nm over F . We may replace L by a definable supplement of H in K, an suppose that L and H are disjoint. If F is finite, so is L. It follows that H equals K, and f is onto. So let us suppose F infinite. Let Γ be a finitely generated algebraic closure containing an F -basis b of L. As L and H are disjoint, we get

Γ = L ∩ Γ ⊕ H ∩ Γ
where L ∩ Γ is a finite dimensional F ∩ Γ-vector space. The intersection I of the sets λH ∩ Γ, where λ runs over Γ is a finite intersection. It is an ideal of Γ. Note that this holds for every finitely generated algebraic closure Γ containing b. If I = Γ for every such Γ, then f is surjective. So we may assume that I is zero for every sufficiently large Γ. But H ∩ Γ has finite F ∩ Γ-codimension, and so has I. It follows that Γ is an algebraic extension of Γ ∩ F . But Γ ∩ F is algebraically closed as Γ and F are. Hence F contains Γ. As this holds for every large enough Γ, the field F and K are equal. If Kerf has F -dimension 1, then f is trivial, a contradiction. So Kerf is zero, and f is onto by Lemma 1.3.

Let K be a small infinite field, together with a definable field morphism σ. We call F the subset of points in K fixed by σ. It is a definable subfield of K hence either finite or algebraically closed. The kernel of σ being an ideal of K, either σ is zero, or it is injective hence surjective by Lemma 1.3.

Lemma 4.2. For all polynomial P with coefficients in K and degree n, the kernel of P (σ) is an F -vector space having dimension at most n.

Proof. Let x 0 , x 1 , . . . , x n be solutions of the equation

σ n (x) + n-1 i=0 a i σ i (x) = 0
and let C(x 0 , x 1 , . . . , x n ) be their Casoratian, defined by

C(x 0 , x 1 , . . . , x n ) = x 0 x 1 • • • x n σ(x 0 ) σ(x 1 ) • • • σ(x n ) . . . . . . . . . . . . σ n (x 0 ) σ n (x 1 ) • • • σ n (x n ) = x 0 x 1 • • • x n σ(x 0 ) σ(x 1 ) • • • σ(x n ) . . . . . . . . . . . . - n-1 i=0 a i σ i (x 0 ) - n-1 i=0 a i σ i (x 1 ) • • • - n-1 i=0 a i σ i (x n ) = 0
So the solutions x 0 , x 1 , . . . , x n are linearly dependent over F according to [6, Lemma II p. 271].

Lemma 4.3. If the set of points fixed by σ is infinite, the set of points fixed by σ and σ n are the same for every natural number n.

Proof. The field F is algebraically closed. If σ n fixes x, σ fixes the symmetric functions of the roots x, σ(x), . . . , σ n-1 (x), hence the roots.

Theorem 4.4. In a small field of positive characteristic, the only definable field morphism the set of points fixed by which is infinite, is the identity.

Proof. Otherwise, the map σ -Id is onto after Proposition 4.1 and Lemma 4.2 so there is some x satisfying σ(x) = x + 1, hence σ p (x) = x + p = x, a contradiction with Lemma 4.3.

Weakly small rings

All the rings considered here are associative. They may neither have a unit nor be abelian. Let R be a ring. An element r is said to be nilpotent of nilexponent n if n is the least natural number with r n = 0. R is nil if there is some natural number n such that every element is nil of nilexponent at most n. Its nilexponent is the least such n. We write R n for the n times Cartesian product of R, and R (n) for the set

{r 1 • • • r n : (r 1 , . . . , r n ) ∈ R n }. The ring R is nilpotent if R (n) is zero for some n.
Its nilpotency class is the least such n. An idempotent is any non-zero element e with e 2 = e. Two idempotents e, f are orthogonal if ef = f e = 0.

Let A be a subset of R. An element r left annihilates A if rA is zero. We write Ann R (A) for the left annihilators of A in R, that is the set of elements in R that left annihilates A. Symmetrically, Ann R (A) will stand for the right annihilator of A. Note that Ann R Ann R Ann R (A) equals Ann R (A).

The characteristic of R is the least non-zero natural number n such that the n times sum r + • • • + r is zero for every r in R. If such a number does not exists R has characteristic zero.

We begin by "dimension 1" rings in the sense of [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes de et corps de dimension un et de déterminer leurs propriétés algébriques[END_REF]. Recall that a d-minimal group is abelian-by-finite [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes de et corps de dimension un et de déterminer leurs propriétés algébriques[END_REF].

Proof. For every set X, the set AnnX is type-definable with parameters in X. So Ann Γ Ann Γ (Γ i ) is Γ-type-definable. The chain Ann Γ Ann Γ (Γ i ) is descending, so Ann Γ Ann Γ (Γ n ) equals Ann Γ Ann Γ (Γ n+1 ) for some natural number n after the weakly small chain condition.

Remark 5.7. Propositions 5.5 and 5.6 are incomparable. The first one is global, with parameters in a definable closure, whereas the second one is local, but with parameters in an algebraic closure. They both hold for chains of right annihilators.

Corollary 5.8. In a weakly small ring R, for every element r, there is a natural number n such that R = r n .R ⊕ Ann R (r n ) Proof. By Proposition 5.5, the chain Ann R (r), Ann R (r 2 ), . . . becomes stationary at some step n. It follows that the right multiplication map by r n is an injective homeomorphism of the group R + /Ann R (r n ). By Lemma 1.3, it must be onto.

5.2. The Jacobson radical. Every abelian group can be given a ring structure with trivial multiplication. Given any ring, one may be willing to isolate its "trivial" part. This is one reason to introduce the Jacobson radical. Among other notions of radical, the one introduced by Jacobson seems to be the more efficient to establish structure theorems for rings with zero radicals : Recall that a ring is right Artinian if every decreasing chain of right ideals is stationary. An element r is right quasi-regular if there is some s such that r + s + rs is zero. We write J(R) the Jacobson radical of R, defined for our purpose by : Whereas most other radicals are definable in second order logic, the Jacobson radical is definable by the following first order formula : ∀y∃z(xy + z + xyz = 0).

Proposition 5.11. In the Jacobson radical of a weakly small ring, the algebraic closure of any finite tuple is nilpotent.

Proof. Let J be the Jacobson radical, and Γ be a finitely generated algebraic closure in J. By Proposition 5.6, there is a natural number n such that Ann Γ (Γ

(n) ) equals Ann Γ (Γ (n+1) ). If Γ is not nilpotent, then Γ \ Ann Γ (Γ (n)
) is not empty. Among the a in Γ \ Ann Γ (Γ (n) ), let us choose one such that the group aJ ∩ Γ be minimal (among the groups {γJ ∩ Γ : γ ∈ Γ}). This is possible by the weakly small chain condition 1.5. Neither aΓ (n) , nor aΓ (n+1) equal zero, so there exists some b in Γ such that abΓ (n) is not zero. We claim that ab does not belong to abJ. Otherwise, there is some c in J with ab = abc. So -c belongs to J too, and there exists some d such that dccd equals zero. It follows that the equality abd = abcd = ab(dc) hold, hence abc is zero. So ab is zero, a contradiction. Thus, one has abJ ∩ Γ < aJ ∩ Γ, a contradiction.

Remark 5.12. Let J be the radical of a small ring. By Proposition 5.11, J is nil.

A compactness argument ensures that its nil exponent must be bounded. Call n the nilexponent of J. Dubnov-Ivanov-Nagata-Higman's Theorem states that "a nil algebra of exponent n over a field of characteristic either zero or a prime number greater than n is nilpotent of class at most 2 n -1". See [START_REF] Jacobson | Structure of rings[END_REF]Jacobson il me semble] for a proof of that. The proof extends naturally to the ring context : " ". By [START_REF] Frank | Quasi-endomorphisms in small stable groups[END_REF], the additive group of J is the direct sum of two subgroups J 0 and J m , where J 0 is additively divisible and J m has additive exponent m. One can easily see that J 0 and J m are ideals, of characteristic zero and m respectively. After Dubnov-Ivanov-Nagata-Higman's Theorem, J 0 is nilpotent. Should m be a prime number greater than n, J m (hence J) would be nilpotent too. Is the radical of a small ring nilpotent?

5.3. Abelian ring with zero radical. Let R be a weakly small abelian ring with zero radical. Note that R has no nilpotent element, except zero.

Lemma 5.13. Let Γ be a finitely generated algebraic closure in R.

(1) If Γ is non-trivial, it has an idempotent element e.

(2) If e is the only idempotent element in Γ, then eΓ is a field.

Proof.

(1) Suppose that Γ is non zero, and let r be in Γ \ {0}. By Corollary 5.8, there is a natural number n such that R equals r n R ⊕ Ann(R n ). In fact, one can show that r n R equals r 2n R so the ring R equals r 3n R ⊕ Ann(R n ). So, there is some a in Ann(r n ) and some b such that r n = r 3n b + a holds. As the sum is direct, note that a and r 3n b must be in Γ. This yields r 2n = r 4n b. As r cannot be nilpotent, r 2n b is a non zero, and idempotent. As r 3n b ∈ Γ, it easily follows that r 2n b ∈ Γ.

(2) If e is the only nilpotent element in Γ, one must have r 2n b = e, so r is invertible in eR. As the inverse of r is unique, it must be algebraic over r and e. So eΓ is a field.

Proposition 5.14. In R, any finitely generated algebraic closure Γ is isomorphic to a finite Cartesian product of fields.

Proof. Suppose that Γ is non zero. Then Γ has a idempotent e 1 by Lemma 5.13 [START_REF] Cherlin | On ℵ 0 -categorical nilrings[END_REF]. We may choose it such that the additive group e 1 R ∩ Γ is minimal among the groups {γR ∩ Γ : γ 2 = γ and γ ∈ Γ \ {0}} by Theorem 1.5. If e 1 has an orthogonal idempotent in Γ, we choose one, say e 2 , such that e 2 R ∩ Γ is minimal. If e 1 , e 2 have a common orthogonal idempotent in Γ, we pick e 3 among the ones such that e 3 R ∩ Γ is minimal. We claim that this process must stop. Otherwise, there would be an infinite chain e 1 , e 2 , . . . of pairwise orthogonal idempotents. The chain (Ann Γ (e 1 + • • • + e n )) n≥1 would be strictly decreasing, a contradiction with the weakly small chain condition 1.5. So let e 1 , . . . , e n be such a family of pairwise orthogonal idempotents, of maximal size n. By maximality of n, the ring Ann Γ (e 1 + • • • + e n ) contains no idempotent. It must be zero by Lemma 5.13 [START_REF] Cherlin | On ℵ 0 -categorical nilrings[END_REF]. If for some i the ring e i Γ should possess another idempotent e i a = e i , as e i ae i and e i a are pairwise orthogonal idempotents, one would have e i aΓ < e i Γ, a contradiction with the choice of e i . So every e i Γ is a field by Lemma 5.13(2).

Remark 5.15. Contrary to omega-stable rings, a small abelian ring with zero Jacobson radical need not necessary be isomorphic to a finite Cartesian product of fields. For instance, an infinite atomless boolean ring is ℵ 0 -categorical [19, Poizat, Théorème 6.21].

5.4. Non abelian rings. Let R be any weakly small ring.

Pb interessant a resoudre : si R sans radical, sous quelle condition Γ est-il sans radical?

Lemma 5.16. Let Γ be a finitely generated algebraic closure in R.

(1) For every r in Γ, there is some a in Γ and n in N such that r n = r n ar n .

(2) If Γ is not nil, it has at least one idempotent.

Proof.

(1) As in the proof of Lemma 5.13, for any a in Γ, one can find some b in Γ such that a n = a 2n b. By Corollary 5.8 and symmetry, Γ equals Γa n ⊕ Ann Γ (a n ) so there are some d in Γ and f in Ann Γ (a n ) with b = da n + f . So a 2n = a 2n da 2n .

(2) If Γ is not nil, one may choose a non nilpotent, hence a 2n d is non zero hence an idempotent element.

Proposition 5.17. For any finitely generated algebraic closure Γ in R, the ring Γ/J(Γ) is right artinian.

Proof. Suppose that Γ is non nil. Then it has an idempotent e 1 by Lemma 5.16 [START_REF] Cherlin | On ℵ 0 -categorical nilrings II[END_REF]. We choose it so that the group e 1 Γ is minimal. If there is some idempotent e 2 with e 1 e 2 = 0 and e 2 e 1 ∈ J(Γ), we choose it such that e 2 Γ is minimal. Should this process not stop, there would be an infinite chain e 1 , e 2 , . . . of such idempotent making the sequence (Ann Γ (e 1 + • • • + e n )) n≥1 strictly decreasing, a contradiction. So let be a maximal chain e 1 , . . . , e n of idempotents in Γ with e i e j = 0 and e j e i ∈ J(Γ), and e i Γ minimal for every i < j. Assume first that there be some natural number i, and some idempotent e i a in e i Γ such that e i ae i = e i . Then e ie i ae i and e i ae i are two orthogonal idempotents in e i Γ, so e i ae i Γ < e i Γ, a contradiction with the choice of e i . So for every i, every idempotent e i a in e i Γ verifies e i ae i = e i . It follows from Lemma 5.16 [START_REF] Cherlin | On ℵ 0 -categorical nilrings[END_REF], that every element in e i Γ is either nil or e i -invertible. Any right ideal of e i Γ must either equal e i Γ, or be a nil ideal. Hence Γ/J(Γ) is right Artinian.

Corollary 3 . 5 .

 35 Let a be some finite tuple in D. The sets acl(a), dcl(a) and a have the same centraliser in D. Proof. The inclusions C(acl(a)) ≤ C(dcl(a)) ≤ C(a) are easy. Conversely, suppose

  be properly descending, and D n ∩ Γ be minimal non commutative. The fields extensions D i ∩ Γ/D i+1 ∩ are finite by Lemma 3.7. As D n+1 ∩ Γ is a field, Γ has finite dimension over its centre, bounded by [Γ : D n+1 ∩ Γ] 2 according to [5, Corollary 2 p.49].

Fact 5 . 9 .

 59 (Wedderburn-Artin [7, Theorem 2.1.7]) A right Artinian ring with zero Jacobson radical is isomorphic to a finite Cartesian product of matrix rings over fields.

Fact 5 .

 5 10. (Jacobson [7, Theorem 1.2.3]) The Jacobson radical of a ring is the unique maximal right ideal the elements of which are right quasi-regular.

  Corollary 5.8 yieldsA = (e 1 + e 2 + • • • + e n )A ⊕ Ann(e 1 + • • • + e n )As the e i are pairwise orthogonal, one hasA = e 1 A ⊕ e 2 A ⊕ • • • ⊕ e n A ⊕ Ann(e 1 + • • • + e n )Note that one has (e i R) ∩ Γ = e i Γ for every i. It follows thatΓ = e 1 Γ ⊕ e 2 Γ ⊕ • • • ⊕ e n Γ ⊕ Ann Γ (e 1 + • • • + e n )

  Corollary 5.8, there is a natural number m withΓ = ((e 1 + • • • + e n ) m )Γ ⊕ Ann Γ ((e 1 + • • • + e n ) m ) If Ann Γ (e 1 +• • •+e n ) hasan idempotent e, then ee i = 0 for all i. Moreover, one can easily verify that e 1 eΓ is a nil right Γ-ideal. If follows from Fact 5.10 that e 1 eΓ is included in J(Γ). This contradicts the maximality of n, hence Ann Γ (e 1 + • • • + e n ) is a nil ideal by Lemma 5.16[START_REF] Cherlin | On ℵ 0 -categorical nilrings II[END_REF]. Note that (e 1 + • • • + e n ) m and e 1 + • • • + e n are equal modulo J(Γ). This yields Γ/J(Γ) = (Γ/J(Γ))e 1 ⊕ • • • ⊕ (Γ/J(Γ))e n

Corollary 5.18. Let Γ be a finitely generated algebraic closure in R. The ring Γ/J(Γ) is isomorphic to a finite Cartesian product matrix rings over fields. Proof. By Proposition 5.17, Γ/J(Γ) is right Artinian. Symmetrically, it must be left Artinian. The conclusion follows from Wedderburn-Artin's Theorem 5.9.

Proposition 5.1. An infinite ring with no definable infinite proper subgroup is a ring with trivial multiplication, or a division ring.

Proof. Let R be such a ring. Any definable group morphism of R is either zero or onto. Let us suppose the multiplication non trivial. There exists some r so that rR equal R. There is some e such that re equal a. So eR equals R. If there exists some x with exx = r, then r 2 is zero, and R is zero. Thus exx must be zero for all x, and e is a left unity. Symmetrically, R has a right unit, which must be e. If sR is zero, then s is zero, so the multiplication by a non-zero element is onto, and R is a division ring.

Corollary 5.2. A d-minimal ring has an ideal of finite index which is a field or a ring with trivial multiplication.

Proof. The ring has a smallest definable additive subgroup of finite index I which is an ideal, with no proper infinite subgroup. If the multiplication is non-trivial, I is a division ring by Proposition 5.1. I has a smallest multiplicative subroup G of finite index, which is abelian by [START_REF] Poizat | Quelques tentatives de définir une notion générale de groupes de et corps de dimension un et de déterminer leurs propriétés algébriques[END_REF]. Its centraliser C I (G) is an infinite division ring, and equal I. So the centre of I is infinite, and equals I. (1) If some element does not left divide zero, R has a left unit.

(2) If R is unitary, an element is left invertible if and only if it is right divisible, and its right and left inverses are the same. (3) An element is a left zero divisor if and only if it is not left invertible.

Proof. (1) Let r be a non left zero divisor. Left multiplication by r is injective, so surjective by Lemma 1.3, and there is some e such that re = r. For all s, r(ess) is zero so es equals s, and e is a left unit. (2) If rs = e holds, then right multiplication by r is injective, hence surjective ; there exists some t so that tr = e. Hence te = trs = es. (3) If r does not left divide zero, then R has a left unit, and left multiplication by r is onto : the pre-image of the left unit is a right inverse of r.

Corollary 5.4. A weakly small ring with no zero divisor is a division ring.

Let us state two chain conditions on ascending chains of annihilators : Proposition 5.5. In a weakly small ring R, there is no properly ascending chain of left annihilators of the kind Ann R (δ

where the sets δ i lie in some finitely generated definable closure δ.

Proof. The chain Ann

Let n be some natural number so that the Cantor rank and degree over δ of Ann R Ann R (δ n ) be minimal. As

Proposition 5.6. In a weakly small ring, there is no properly ascending chain of annihilators Ann Γ (Γ 1 ) ≤ Ann Γ (Γ 2 ) ≤ • • • ≤ Ann Γ (Γ i ) ≤ • • • , where the sets Γ i lie in some finitely generated algebraic closure Γ.


Cédric Édric Milliet

SMALL SKEW FIELDS

Keywords: 2000 Mathematics Subject Classification. 03C15, 03C50, 03C60, 12E15 Smallness, skew fields

come

Remark 1.5. Let H < G be A-definable small groups with H ∩ dcl(A) < G ∩ dcl(A). Then, either CB A (H) < CB A (G), or CB A (H) = CB A (G) and dCB A (H) < dCB A (G).

Corollary 1.6. A small integral domain with unity is a field.

Proof. Let R be this ring. If r is not invertible, then 1 / ∈ rR hence rR ∩ dcl(r) R∩dcl(r). Apply Remark 1.5, but R and rR have same rank and degree over r.

Note that R need not have a unity (see Corollary 1.10). More generally, if ϕ is a definable bijection between two definable groups A ≤ B in a small structure, then A equals B.

Proposition 1.7. (Descending Chain Condition) Let G be a small group and g a finite tuple in G. Set H = g or H = dcl(g). In H, there is no strictly decreasing infinite chain of subgroups of the form

G 0 ∩ H > G 1 ∩ H > G 2 ∩ H > • • • , where the G i are H-definable subgroups of G.
Proof. By Remark 5, either the rank or the degree decreases at each step.

Corollary 1.8. Let G be a small group, H < G a finitely generated subgroup of G, and (G i ) i∈I a family of H-definable subgroups of G. There is a finite subset

I 0 ⊂ I such that i∈I G i ∩ H = i∈I0 G i ∩ H.

Another chain condition on images of endomorphisms :

Proposition 1.9. Let G be a small group and h a group homomorphism of G. There exists some integer n such that Imh n equals Imh n+1 .

Proof. Suppose that the chain (Imh n ) n≥1 be strictly decreasing. Consider the following tree G(x)

Imh P P P P P P Imh 2 h(x).Imh 2 ! ! ! ! a a a a ! ! ! ! a a a a Im.h 3 h 2 (x).Imh 3 h(x).Imh 3 h(x).h 2 (x).Imh 3 . . . . . . . . . . . . Consider the partial type Φ(x) := {x / ∈ h -n Imh n+1 , n ≥ 1}.
The sequence (h -n Imh n+1 ) n≥1 is increasing, and each set G \ h -n Imh n+1 is non-empty, so Φ is finitely consistent. Let b be a realization of Φ in a saturated model. The graph G(b) has 2 ω consistent branches, whence S 1 (b) has cardinal 2 ω , a contradiction with G being small. Corollary 1.10. Let G be a small group and h a group homomorphism of G. There exists some integer n such that G equals Kerh n • Imh n .

Proof. Take n as in Proposition 1.9, and set f = h n . We have Imf 2 = Imf , so for all g ∈ G there exists g ′ such that f (g) = f 2 (g ′ ). Hence f (gf (g ′ ) -1 ) = 1 and gf

(g ′ ) -1 ∈ Kerf , that is g ∈ Kerf • Imf .
It was shown in [START_REF] Wagner | Small stable groups and generics[END_REF] that a definable injective homomorphism of a small group is surjective. Note that this follows again from Corollary 1.10.

Small skew fields

Recall a result proved in [START_REF] Wagner | Small fields[END_REF] : Fact 2.1. An infinite small field is algebraically closed.

Let D be an infinite small skew field. We begin by analysing elements of finite order.

Lemma 2.2. Let a ∈ D an element of order n < ω. Then a is central in D. Proof. Either D has zero characteristic, so Z(D) is infinite, hence algebraically closed. But Z(D)(a) is an extension of Z(D) of degree d ≤ n, whence a ∈ Z(D).
Or D has positive characteristic. Suppose that a is not central, then [3, Lemma 3.1.1 p.70] there exists x in D such that xax -1 = a i = a. If x has finite order, then all elements in the multiplicative group x, a have finite order. Hence x, a is commutative [3, Lemma 3.1.3 p.72], contradicting xax -1 = a. So x has infinite order. Conjugating m times by x, we get x m ax -m = a i m . But a and a i have same order n, with gcd(n, i Let us now show that a small skew field is connected, that is to say, has no definable proper subgroup of finite index.

) = 1. Put m = φ(n). By Fermat's Theorem, i m ≡ 1[n], so x m and a commute. Then L = Z(C D (a, x m )) is a definable infinite commutative subfield of D which contains a. Let L x be {l ∈ L, x -1 lx = l}. This is a proper subfield of L. Moreover 1 < [L x (a) : L x ] ≤ n. But L x is infinite as it contains x. By Fact 2.
Proof. Clearly C D (a) ≤ C D (a n ). Consider L = Z(C D (a n )). It is algebraically closed by Fact 2.1, but L(a) is a finite commutative extension of L, whence a ∈ L and C D (a n ) ≤ C D (a).
Now suppose that D is not commutative. We shall look for a commutative centralizer C and show that the dimension [D : C] is finite. This will yield a contradiction.

Lemma 2.7. Let a ∈ D, t ∈ D \ im(x → ax -xa) and ϕ : x → t -1 • (ax -xa). Then D = imϕ ⊕ kerϕ. Moreover, if k = dcl(a, t, x), where x is a finite tuple, then k = imϕ ∩ k ⊕ kerϕ ∩ k. Proof. Let K = kerϕ = C D (a). Put I = imϕ; this is a right K-vector space, so I ∩ K = {0}, since 1 ∈ K ∩ I is impossible by the choice of t. Consider the morphism φ : D + /K -→ D + /K x -→ ϕ(x)
φ is an embedding, and D + /K is small ; by Corollary 10, φ is surjective and D/K = φ(D/K), hence D = I ⊕ K. Now, let k = dcl(a, t, x) where x is a finite tuple of parameters in D. I and K are k-definable. For all α ∈ k there exists a unique couple (α 1 , α 2 ) ∈ I × K such that α = α 1 + α 2 , so α 1 and α 2 belong to dcl(α, a, t) ≤ k, that is to say

k = I ∩ k ⊕ K ∩ k.
Lemma 2.8. For every a / ∈ Z(D), the map ϕ a : x → ax -xa is onto.

Proof. Suppose ϕ a not surjective. Let t / ∈ imϕ a , and k = dcl(t, a, x) be a non commutative subfield of D for some finite tuple x. Consider the morphism ϕ : Theorem 2.9. A small field in non-zero characteristic is commutative.

D + -→ D + x -→ t -1 .(ax -xa) Set H = imϕ, and K = C D (a) = kerϕ. By Lemma 2.7 we get k = H ∩ k ⊕ K ∩ k. Let N = I a i H be a finite intersection of left-translates of H by elements in k, such that N ∩ k be minimal. We have N ∩ k = i∈I a i H ∩ k = d∈k dH ∩ k, so N ∩k is a left ideal. Moreover, H ∩k is a right K ∩k vector-space of codimension 1. Then N ∩ k has codimension at most n =| I |. If N ∩ k = k, then H ∩ k = k, whence K ∩ k = {0}, a contradiction. So N ∩ k is trivial and, k is a K ∩ k-
Proof. Let a ∈ D be non-central, and let us show that x → ax-xa is not surjective. Otherwise there exists x such that ax -xa = a, hence axa -1 = x + 1. We would then have a p xa -p = x + p = x, and x ∈ C D (a p ) \ C D (a), a contradiction with Lemma 2.2.

3. Open problems Note that a positive answer to question 3 implies a positive answer to question 4, as all the proves given still hold. In general, one can prove divisibility and connectivity of an infinite weakly small field. So D × is divisible and the proof of Proposition 2.5 still holds.

Proposition 3.3. An infinite weakly small field is connected.

  1, it is algebraically closed and cannot have a proper extension of finite degree. Proposition 2.3. Every element of D has a n th root for each n ∈ ω. Proof. Let a ∈ D. If a has infinite order, Z(C D (a)) is an infinite commutative definable subfield of D. Hence it is algebraically closed, and a has an n th root in Z(C D (a)). Otherwise a has finite order. According to Lemma 2.2 it is central in D. Let x ∈ D have infinite order. Then a ∈ Z(C D (a, x)), a commutative, infinite, definable, and thus algebraically closed field. Remark 2.4. Note that since D × is divisible, it has elements of arbitrary large finite order, which are central by Lemma 2.2. Taking D omega-saturated, we can suppose Z(D) infinite.

Proposition 2 . 5 .

 25 D is connected. Proof. Multiplicatively : By Proposition 2.3, D × is divisible so has no subgroup of finite index. Additively : Let H be a definable subgroup of D + of finite index n. In zero characteristic, D + is divisible, so n = 1. In general, let k be an infinite finitely generated subfield of D. Consider a finite intersection G = i∈I d i H of left translates of H by elements in k such that G ∩ k is minimal ; this exists by the chain condition. By minimality, G ∩ k = d∈k dH ∩ k, so G ∩ k is a left ideal of k. Furthermore, G is a finite intersection of subgroups of finite index in D + ; it has therefore finite index in D. Thus G ∩ k has finite index in D ∩ k = k, and cannot be trivial, so G ∩ k = k = H ∩ k. This holds for every infinite finitely generated k, whence H = D. Now we look at elements of infinite order. Lemma 2.6. a ∈ D have infinite order. Then C D (a) = C D (a n ) for all n > 0.

  vector space of dimension at most n. By [2, Corollary 2 p.49] we get [k :K ∩ k] = [Z(k)(a) : Z(k)]. But Z(k) = Z(C D (k)) ∩ k with Z(C D (k))algebraically closed. Note that every element of k commutes with Z(C D (k)), so a ∈ Z(k), which is absurd if we add b / ∈ C D (a) in k.

Proposition 3 . 2 .

 32 Every element in an infinite weakly small field D has a n th root for all n ∈ ω.Proof. Let a ∈ D. In zero characteristic, Z(C D (a)) is an infinite definable commutative subfield of D, hence weakly small. According to [5, Proposition 9], every element in Z(C D (a)) has a n th root. In positive characteristic, we can reason as in the proof of Lemma 12, and find y with infinite order which commutes with a. Apply one more time[START_REF] Wagner | Small fields[END_REF] Proposition 9] to Z(C D (a, y)).

  3.1. Zero characteristic. Note that we just use characteristic p in proof of theorem 19 to show that there exist a / ∈ Z(D) such that x → ax -xa is not surjective. Thus questions 1 and 2 are equivalent : Question 1. Is a small skew field D of zero characteristic commutative ? Question 2. Is every x → ax -xa surjective onto D for a /

∈ Z(D) ? 3.2. Weakly small fields. small structures have been introduced to give a common generalization of small and minimal structures. Minimal fields are known to be commutative. Definition 3.1. A structure M is weakly small if for all finite set of parameters A in M , there are only countably many 1-types over A. Question 3. Is a weakly small field algebraically closed ? Question 4. Is a weakly small skew field commutative ?


Kate M Fleming
email: kate.fleming@nottingham.ac.uk

Consultant Guruprasad P Aithal

Research Associate

Hepatobiliary Physician

Clinical Associate Professor Tim R Card

Clinical Associate Professor Joe West

and clinical progression of disease in people with cirrhosis: a cohort study

Keywords: cirrhosis, decompensation, epidemiology, progression

HAL is

Aim

To determine the rate of decompensation and clinical progression of disease in patients with cirrhosis based upon clinical symptoms recorded electronically in general practice data.

Methods

Using Cox proportional hazards regression we modelled the rate of decompensation for patients from the UK General Practice Research Database with a diagnosis of cirrhosis between 1987 and 2002. We determined the clinical progression in the first year following diagnosis and subsequently categorising patients through time according to a simple clinical staging system agreed at the Baveno IV consensus conference.

Results

The rate of decompensation in patients with compensated cirrhosis was found to be 11% overall. The rate of decompensation was higher in the first year (at 31% compared with 7.3% afterwards) and in patients with an alcoholic aetiology. Patients with compensated cirrhosis had a one-year probability of proceeding directly to death of 7% compared with 20% in patients with decompensated cirrhosis. Using data recorded in general practice records it has been possible to determine the rate of decompensation and the clinical progression of disease in people with cirrhosis.

Conclusions

Deleted: The rate of decompensation was rapid at 31% in the first year following diagnosis.

Introduction

The natural history of cirrhosis has long been of interest to gastroenterologists and hepatologists [START_REF] Macdonald | The natural history of postnecrotic cirrhosis; a study of 221 autopsy cases[END_REF] [2] [3] [4] [START_REF] Powell | Cirrhosis of the liver. A comparative study of the four major aetiological groups[END_REF] [6] [START_REF] Prytz | Morbidity, Mortality, and incidence of cirrhosis in Denmark 1976-1978[END_REF] as knowledge of the progression of the disease allows planning of management and development of effective strategies for the primary prevention of complications and is of obvious interest to patients who wish to understand their risk of death, distressing symptoms and time to referral for consideration for transplantation. While in recent years advances have been made in the management of portal hypertension [START_REF] Sanyal | Portal hypertension and its complications[END_REF] with a consequent apparent improvement in some outcomes [START_REF] Stokkeland | Improved prognosis for patients hospitalized with esophageal varices in Sweden 1969-2002[END_REF] we have surprisingly little information on the effect of these developments on the natural history of cirrhosis in a general population-based setting. The most commonly referenced studies have focussed on those patients who are admitted to hospital. In addition, while scoring systems such as Child-Pugh, MELD, UKELD enable the ranking of patients at an individual level and near the end stage of disease, we cannot at present provide useful absolute estimates of the rate of decompensation, clinical progression or mortality to patients in the earlier stages of disease.

In a recent comprehensive systematic review of the natural history and prognostic indicators of survival in cirrhosis which included 118 pertinent publications [START_REF] D'amico | Natural history and prognostic indicators of survival in cirrhosis: a systematic review of 118 studies[END_REF] the authors reported a wide range of estimates of survival and factors that predicted death but many methodological difficulties were also described. Perhaps most saliently they provided for the first time one year outcome probabilities for cirrhosis according to the clinical stages of disease agreed at the Baveno IV consensus conference [START_REF] De Franchis | Evolving consensus in portal hypertension. Report of the Baveno IV consensus workshop on methodology of diagnosis and therapy in portal hypertension[END_REF]. However, as their estimates were based on the experience in a sole hospital in Sicily in the late 1970s-early1980s they may not be widely generalisable. We therefore do not have contemporary estimates of the progression of disease in cirrhosis at a population level.

Increasingly people are being diagnosed with cirrhosis at earlier stages of the disease i.e. compensated rather than decompensated. We have consequently taken the opportunity presented by prospectively-recorded electronic primary care data from the UK to determine

Methods

Description of GPRD

The General Practice Research Database (GPRD) is a longitudinal database consisting of anonymous computerised general practice records for over 13 million patients in the UK, including over 50 million patient-years of usable data. Practices are subjected to regular data quality checks and audits to maintain 95% inclusion of prescribing and morbidity events. Data contained within this database are recorded through direct entry during general practice appointments and following communication from secondary care. Data are coded based on both the Oxmis and Read medical coding dictionaries. The GPRD has previously been shown to be broadly representative of the population of the UK [START_REF] Walley | The UK General Practice Research Database[END_REF] of which it represents approximately a 4-6% sample. Approval for this study was given by the Scientific and Ethical Committee of the GPRD.

Study population

We obtained all records of patients with a diagnostic code for any liver disease within the GPRD between June 1987 and April 2002. Patients aged 25 or over were selected based on the presence of a diagnostic or therapeutic code for cirrhosis, oesophageal varices and/or portal hypertension to represent a cohort of adult diagnosed cirrhosis. This age cut-off was designed to avoid including patients who may have had the onset of cirrhosis under the age of 18. Each patient was assigned a date of diagnosis of cirrhosis as the date of the first record of any of these codes.

Definitions

We then categorised patients with cirrhosis through time following date of diagnosis according to the four stages of cirrhosis as agreed at the Baveno IV conference [START_REF] De Franchis | Evolving consensus in portal hypertension. Report of the Baveno IV consensus workshop on methodology of diagnosis and therapy in portal hypertension[END_REF] with GI bleeding, with or without ascites. Stages 1 and 2 together correspond to compensated cirrhosis with stages 3 and 4 together corresponding to decompensated cirrhosis. Each patient was assigned up to four dates corresponding to the earliest recording of the symptoms representing each stage. To minimise the potential for misclassification of patients with decompensated cirrhosis as patients with compensated disease we additionally looked for the recording of one or more prescriptions for spironolactone as evidence of decompensation.

Death was defined using a combination of the patient's registration status within the GPRD and medical codes for death, with the earliest date of these being assigned as the date of death.

Presumed aetiology of cirrhosis was defined as alcoholic, viral hepatitis, autoimmune liver disease or metabolic liver disease as previously reported. [START_REF] Fleming | Incidence and prevalence of cirrhosis in the United Kingdom, 1992-2001: A general population-based study[END_REF] For the purposes of analyses all cases not defined as alcoholic were classified as non-alcohol related cirrhosis.

Statistical analyses

Using Cox proportional hazards we modelled the rate of decompensation in the group of patients with compensated disease. Subjects entered the analysis period at the date of earliest recorded code for stage 1 or 2 and exited at the earliest of date of earliest recorded code for stage 3 or 4, date of death, date of liver transplant, date of deregistration with their general practice or 30 April 2002 which was the last date of available data in this dataset.

We examined the rate of decompensation for subjects with alcoholic cirrhosis and those with non-alcohol related cirrhosis. Follow-up time was split at 1 year and we modelled the rate of decompensation during the first year after diagnosis and subsequent to that first year again stratifying by presumed aetiology.

We examined the probabilities of patients with cirrhosis progressing from an individual stage to a subsequent stage or death within one year and across the whole time period of their records with 95% confidence intervals being calculated assuming a binomial distribution.

Deleted: We defined patients as having underlying alcoholic cirrhosis if there was any mention in their GPRD record of alcoholism, alcohol abuse, addition or dependence, or 'problem drinking'. We also included in this group patients who had a code for referral to alcohol cessation services.

Deleted: All other cases of cirrhosis

Results

A total of 4,537 patients were identified with cirrhosis. Table 1 shows selected clinical and demographic information about the study population. 56.1% entered the study cohort at stage 1, 12.8% at stage 2, 23.0% at stage 3 and 8.1% at stage 4. A total of 2307 (50.9%) of cirrhosis cases were classified as alcoholic cirrhosis with the remainder being assigned as non-alcohol related cirrhosis for the purposes of the subsequent analyses.

Rate of decompensation

Overall, the rate of decompensation for those patients with compensated disease was 11.8% per year adjusted for age and sex (95%CI[11.2%, 12.5%]). This figure varied considerably both by length of follow-up and by presumed aetiology. A Kaplan-Meier survival curve showing the difference in rate of decompensation by presumed aetiology is shown in figure 1.

During the first year after diagnosis the rate of decompensation was 31.0% adjusted for age and sex (95%CI [28.8%, 33.4%]); for those patients with a presumed aetiology of alcoholic cirrhosis the rate of decompensation during the first year was 37.6% (95%CI [34.1%, 41.5%]) compared with 25.2% (95%CI [22.6%, 28.2%] for those with non-alcohol related cirrhosis.

Following the first year the rate of decompensation did not vary much by presumed aetiology with a rate of 7.3% per year adjusted for age and sex (95%CI [6.5%, 8.2%]) for those with alcoholic cirrhosis compared with an adjusted rate of decompensation of 5.5% per year (95%CI [4.8%, 6.2%)] for those with non-alcohol related cirrhosis.

Progression

Outcome probabilities in the first year are shown in figure 2. Patients in stage 1 and 2 had a one-year probability of proceeding directly to death of around 7%. The probability of progression to death within 1 year was highest for patients within stage 3, rather than stage 4 (at 20.1% and 18.2% respectively) though this difference was not statistically significant. The probabilities of progressing both during the first year and state progressions subsequently for those with alcoholic cirrhosis and non alcohol-related cirrhosis are shown in table 2 and 3 respectively. Reading across the table shows the stage at which patients began their follow-up. The rows in the column represent the stage to which the patients directly progressed. For example looking at the first column we can see that alcoholics who began follow-up in stage 1 had a 64.6% chance of remaining in stage 1 during the first year, with 2.9% progressing directly to stage 2 (oesophageal varices), and so on. In the first year following diagnosis patients with alcoholic cirrhosis had a slightly higher probability of progressing to another stage of cirrhosis than those with non-alcohol related cirrhosis but a lower probability of progressing directly to death. Across the whole study period patients with a presumed aetiology of alcoholic cirrhosis were again more likely to progress to another later stage of cirrhosis than those with non-alcohol related cirrhosis. The differences in progression directly to death were not so marked across the whole time period, rather less patients with alcoholic cirrhosis remained in their entry stage with the exception of patients in stage 3.

Discussion

Using the clinical observations recorded in primary care records of people with cirrhosis we have described the progression of disease as experienced by patients and as recorded by primary care physicians in the UK. Progression of disease is rapid with the rate of decompensation in patients with compensated cirrhosis being 11% per year, but is particularly rapid in the first year following diagnosis at 31% in this first year. We have used a relatively simple staging system, based on the presence or absence of symptoms rather than relying on laboratory tests, mimicking the Baveno IV consensus clinical staging system to classify patients with cirrhosis. The classification is useful in long-term planning of management of patients with cirrhosis. We have applied this classification to an ambulatory population from primary care to describe the natural history of this disease. For example, in those patients without recorded evidence of portal hypertension, i.e. those in stage 1 of disease, there was about a 11% probability of progressing to a more severe form of disease (development of ascites, varices or GI bleeding) within a year alongside an 7% likelihood of dying. In those patients with stage 2 disease, the chance of developing ascites or bleeding was approximately 24% in the following year alongside a 7% risk of death. By virtue of the electronic primary care data from a broad sample of primary care physicians at our disposal we have a constructed a large, representative, population-based cohort of patients with cirrhosis. This cohort was identified reasonably recently (1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002) and therefore the results we have generated reflect the natural history of cirrhosis during this period. As this is a population based cohort, it is unlikely to have been affected by the variation in referrals and follow-ups seen in cohorts selected from secondary care. We are confident based upon our own previous validation study (which showed the vast majority of patients with a recorded code for cirrhosis had available extra evidence from secondary care [START_REF] Fleming | Incidence and prevalence of cirrhosis in the United Kingdom, 1992-2001: A general population-based study[END_REF] that for the diagnosis of cirrhosis in general this coding within primary care is good. The additional clinical signs and symptoms of decompensation may not always be recorded as accurately however, unless they are of obvious clinical relevance to the GP. Hence ascites is probably symptomatic and hence, likely to be moderate to large-volume, clinically significant ascites (rather than that only identified by ultrasound). We acknowledge there may have been some misclassification between stages 1 and 2 as the recognition of development of oesophageal varices in the absence of bleeding depends on endoscopic surveillance which, for the period our data covers, was not necessarily uniform in the UK.

We believe though that it is unlikely that much misclassification has occurred between the compensated and decompensated disease states nor between stages 3 and 4 as events such as bleeding oesophageal varices are of such importance that they are highly unlikely to be inaccurately recorded.

For the purposes of predicting progression of disease and mortality we were unable to calculate the typically used Child-Pugh classification [START_REF] Child | Surgery and portal hypertension[END_REF][START_REF] Pugh | Transection of the oesophagus for bleeding oesophageal varices[END_REF] or the more recently derived model for end-stage liver disease (MELD) [START_REF] Kamath | A model to predict survival in patients with end-stage liver disease[END_REF] as we had no measures of blood indices available. However, we argue that the strength of Baveno IV consensus clinical staging system is that it allows such a broad evaluation of the natural history of patients with cirrhosis without relying upon laboratory tests. In addition, unlike Child-Pugh and MELD scores which are of limited value in early stages of cirrhosis where specific laboratory tests are unaltered (e.g. Child-Pugh class A patients), the Baveno classification can be applied to both early and late stage disease. This not only allows us to describe disease progression, but also to do so in a way which is intelligible both to non-specialist clinicians, and to patients. Of course, the simplicity of the Baveno classification is also one of its weaknesses in that many other factors that are known to influence the progression of cirrhosis, such as the development of hepatocellular carcinoma, hepatic encephalopathy and nutritional status are not included within the criteria.

We have included both incident and prevalent cases of cirrhosis by design as there is clear evidence that cirrhosis, as with most chronic diseases, has a wide heterogeneity of presentation. The results of our study therefore reflect the real world of clinical practice within the general population which is unlikely to be true of series collected from within secondary care centres with a particular interest in the disease. This may explain some of the discrepancies with previously reported data as discussed below.

In comparison with historical studies of the natural history of cirrhosis [START_REF] Macdonald | The natural history of postnecrotic cirrhosis; a study of 221 autopsy cases[END_REF] [START_REF] D'amico | Natural history and prognostic indicators of survival in cirrhosis: a systematic review of 118 studies[END_REF] our work shows that even in the later stages of the 20th century progression of disease is rapid. Following our replication of D'Amico et al's [START_REF] D'amico | Natural history and prognostic indicators of survival in cirrhosis: a systematic review of 118 studies[END_REF] outcome model using the Baveno IV stage of disease system we found both similarities and differences with their reported findings. We found a similar clear demarcation of risk of death between compensated disease (stage 1 and stage 2) and decompensated disease (stage 3 and 4) although the probability of death within one year we observed for stage 1 and stage 2 disease was substantially higher than they reported. In contrast, we did not find a difference in the rate of progression to death for those patients with ascites rather than bleeding oesophageal varices. The explanation for these disparities most likely lies in the differences in methodology and population selection we have used. In the current study we have taken each individual's record and defined their progression of disease within 1 year, whereas D'Amico et al appears to report outcome probability estimates which were averaged risks from 10 years of follow up. In addition we have used a population-based approach to collection of cases rather than the single centre approach previously used. The great benefits of a population based approach i.e. avoiding bias and improving generalisability need no rehearsal, but in this case the potential for misclassification of minor degrees of decompensation consequent upon lack of direct access to cases may explain some of the difference in our results. It is likewise probable that much of the difference between our results is a consequence of the fact that the UK, with low levels of chronic viral hepatitis, has a different aetiological picture with respect to cirrhosis to that seen in Sicily (the setting from which the Baveno classification was derived). Finally some of the difference may be due to the era of data collection. Our finding that ascites is as severe a marker of poor prognosis as bleeding fits with the prevailing view that since the late 1970s widely available important improvements have been made in the management and outcomes of variceal bleeding [START_REF] Garcia-Tsao | Management of varices and variceal hemorrhage in cirrhosis[END_REF] with improvements in the treatments for ascites broadly limited to the administration of antibiotics in cases of spontaneous bacterial peritonitis which is a much less frequent complication than bleeding.

[2] [3] [4] [5] [6]
A recent paper from Denmark looking at only alcoholic cirrhosis has provided similarly greater estimates for the 1-year progression directly to death in patients with no reported complications of 10% and similar estimates for 1-year progression directly to death for patients with clinical symptoms of ascites and variceal bleeding. [START_REF] Jepsen | Clinical course of alcoholic liver cirrhosis: A Danish population-based cohort study[END_REF] There are no such comparable figures for patients with non-alcohol related cirrhosis available in current published literature.

In summary, our study has described the disease progression following a diagnosis of cirrhosis in contemporary clinical practice using data recorded electronically in general practice. Our reproduction of the use of a simple, internationally recognised, Baveno IV consensus clinical staging system is practical, useful and relevant to patients with cirrhosis and clinicians in the UK and beyond (within the limits of the assumptions underpinning its design). Of particular interest is the overall rate of decompensation for all those with cirrhosis as a contemporary estimate of this has not been readily available. Clearly further validation of the approach we have used to stage disease in other datasets will be of value.

We conclude that our results in conjunction with the previously observed increasing incidence of cirrhosis emphasise the growing threat of liver disease to public health.

F o r P e e r R e v i e w

  population-based estimates of the rate of decompensation in people with compensated cirrhosis as well as estimates of the manner in which the disease progresses once identified.

  decompensation and the progression of disease in patients with cirrhosis based upon the Baveno IV stage of disease criteria.

  We again stratified the cohort into those with alcoholic cirrhosis and those with non-alcohol related cirrhosis. Stata version 9.2 SE was used for all statistical analyses.

  State progressions across the whole study period are shown in figure3Of note is the high proportion of people in the early stages of disease i.e. patients with compensated cirrhosis, who progressed directly to death without the recording of other clinical symptoms of decompensation. These results support the categorisation of stages 1 and 2 together as representing compensated disease and stages 3 and 4 together as representing decompensated disease.

Deleted: 2 .

 2

Figure 3 .

 3 Figure 3. Probabilities for progression across whole record, all patients

Table 1 .

 1 Selected demographic and clinical characteristics of study population

	Deleted: while there have been no
	interventions that have markedly
	improved survival in patients with
	ascites.
	Deleted: ,

Figure 1. Kaplan-Meier survival estimates for decompensation in subjects with compensated alcoholic and non-alcohol related cirrhosis

	0.00 0.25 0.50 0.75 1.00 Non-alcohol related cirrhosis Proportion remaining in compensated state Alcoholic cirrhosis Number at risk F 1493 Kaplan-Meier survival estimates 665 423 260 1630 756 446 276 0 2 4 6 Follow-up time (years) Alcoholic cirrhosis Non-alcohol related cirrhosis o r P e Stage 1 No ascites No varices Stage 2 Varices No ascites Ascites +/-varices Stage 4 GI bleeding +/-ascites 3.1% 5.6% 177 171 8 6.6% 17.2% 20.1% 6.6% 7.2% 1.1% F o r P e e r R e v i e w Death Stage 3 18.2% 20.3%
		e r
	Figure 2.	R
		e
		v i e
		w

Probabilities for progression within one year, all patients

			15.5%
	Stage 1
	No ascites			Death
	No varices
			6.0%	13.6%
	Stage 2
	Varices
	No ascites			42.9%
			32.5%
	Stage 3 Ascites	31.4%		45.4%
	+/-varices
				12.1%
	Stage 4 GI bleeding	3.4%	10.2%
	+/-ascites

F o r P e e r R e v i e w

Table 2 . Probability of progression in first year, patients with alcoholic and non-alcohol related cirrhosis

 2

	Stage from

Table 3 . State progression across whole record, patients with alcoholic and non-alcohol related cirrhosis

 3

	Stage from

There are no conflicts of interest to declare for any of the authors. The funders of this study had no role in the design, analysis or interpretation of the data. JW and KMF are funded by a National Institute for Health Research Clinician Scientist Fellowship awarded to JW.