Walid Al-Kawarit
email: walid.al-kawarit@unicaen.fr

Franc ¸ois Couchot
email: francois.couchot@unicaen.fr

COMPARISON OF SOME PURITIES, FLATNESSES AND INJECTIVITIES

Keywords: 2000 Mathematics Subject Classification. Primary 16D40, 16D50, 16D80. Key words and phrases. (n, m)-pure submodule, (n, m)-flat module, (n, m)-injective module, (n, m)-coherent ring

In this paper, we compare (n, m)-purities for different pairs of positive integers (n, m). When R is a commutative ring, these purities are not equivalent if R doesn't satisfy the following property: there exists a positive integer p such that, for each maximal ideal P , every finitely generated ideal of R P is p-generated. When this property holds, then the (n, m)-purity and the (n, m ′ )-purity are equivalent if m and m ′ are integers ≥ np. These results are obtained by a generalization of Warfield's methods. There are also some interesting results when R is a semiperfect strongly π-regular ring. We also compare (n, m)-flatnesses and (n, m)-injectivities for different pairs of positive integers (n, m). In particular, if R is right perfect and right self (ℵ 0 , 1)-injective, then each (1, 1)-flat right R-module is projective. In several cases, for each positive integer p, all (n, p)-flatnesses are equivalent. But there are some examples where the (1, p)-flatness is not equivalent to the (1, p + 1)-flatness.

All rings in this paper are associative with unity, and all modules are unital. Let n and m be two positive integers. A right R-module M is said to be (n, m)presented if it is the factor module of a free right module of rank n modulo a m-generated submodule. A short exact sequence (Σ) of left R-modules is called (n, m)-pure if it remains exact when tensoring it with any (n, m)-presented right module. We say that (Σ) is (ℵ 0 , m)-pure exact (respectively (n, ℵ 0 )-pure exact if, for each positive integer n (respectively m) (Σ) is (n, m)-pure exact. Let us observe that the (1, 1)-pure exact sequences are exactly the RD-exact sequences (see [START_REF] Warfield | Purity and algebraic compactness for modules[END_REF]) and the (ℵ 0 , ℵ 0 )-exact sequences are the pure-exact sequences in the Cohn's sense. Similar results as in the classical theories of purity hold, with similar proofs. In particular, a left R-module is (n, m)-pure-projective if and only if it is a summand of a direct sum of (m, n)-presented left modules, and each left R-module has a (n, m)-pure-injective hull which is unique up to an isomorphism.

In this paper, we compare (n, m)-purities for different pairs of positive integers (n, m). When R is commutative, we shall see that some of these purities are equivalent only if R satisfies the following property: there exists a positive integer p such that, for each maximal ideal P , every finitely generated ideal of R P is pgenerated. When this property holds, then (n, m)-purity and (n, m ′ )-purity are equivalent if m and m ′ are integers ≥ np. These results are obtained by using the following: if R is a local commutative ring for which there exists a (p + 1)-generated ideal, where p is a positive integer, then, for each positive integer n, for each integer m, n(p -1) + 1 ≤ m ≤ np + 1, there exists a (n, m)-presented R-module whose endomorphism ring is local. It is a generalization of the Warfield's construction of indecomposable finitely presented modules when R is not a valuation ring.

When R is semiperfect and strongly π-regular, we show that there exists an integer m > 0 such that, for any integer n > 0, each (n, m)-pure exact sequence of right modules is (n, ℵ 0 )-pure exact if and only if there exists an integer p > 0 such that every finitely generated left ideal is p-generated.

As in [START_REF] Zhang | On (m, n)-injective modules and (m, n)-coherent rings[END_REF] we define (n, m)-flat modules and (n, m)-injective modules. We also compare (n, m)-flatnesses and (n, m)-injectivities for different pairs of positive integers (n, m). In particular, if R is a right perfect ring which is right self (ℵ 0 , 1)injective, then each (1, 1)-flat right R-module is projective. For many classes of rings, for each positive integer p, we show that the (1, p)-flatness implies the (ℵ 0 , p)flatness, but we have no general result. If R is a local commutative ring with a non finitely generated maximal ideal P satisfying P 2 = 0, then for each positive integer p, there exists an R-module which is (ℵ 0 , p)-flat (resp, (ℵ 0 , p)-injective) and which is not (1, p + 1)-flat (resp, (1, p + 1)-injective).

As in [START_REF] Zhang | On (m, n)-injective modules and (m, n)-coherent rings[END_REF] we define left (n, m)-coherent rings. When R is a commutative locally perfect ring which is (1, 1)-coherent and self (1, 1)-injective, we show that R is an IF-ring, each (1, 1)-flat R-module is flat and each (1, 1)-injective R-module is FPinjective. For other classes of rings, for each positive integer p, we show that the left (1, p)-coherence implies the left (ℵ 0 , p)-coherence, but we have no general result. If R = V [[X]], the power series ring over a valuation domain V whose order group is not isomorphic to R, then R is a (ℵ 0 , 1)-coherent ring which is not (1, 2)-coherent.

(n, m)-pure exact sequences

By using a standard technique, (see for instance [4, Chapter I, Section 8]), we can prove the following theorem, and similar results hold if we replace n or m with ℵ 0 . Theorem 1.1. Assume that R is an algebra over a commutative ring S and E is an injective S-cogenerator. Then, for each exact sequence (Σ) of left R-modules 0 → A → B → C → 0, the following conditions are equivalent:

(1) (Σ) is (n, m)-pure;

(2) for each (m, n)-presented left module G the sequence Hom R (G, (Σ)) is exact;

(3) every system of n equations over

A m j=1 r i,j x j = a i ∈ A (i = 1, . . . , n)
with coefficients r i,j ∈ R and unknowns x 1 , . . . , x m has a solution in A whenever it is solvable in B;

(4) the exact sequence of right R-modules Hom S ((Σ), E) is (m, n)-pure. Propositions 1.2 and 1.3 can be deduced from [13, Theorem 1]. A left R-module G is called (n, m)-pure-projective if for each (n, m)-pure exact sequence 0 → A → B → C → 0 the sequence 0 → Hom R (G, A) → Hom R (G, B) → Hom R (G, C) → 0
is exact. Similar definitions can be given by replacing n or m by ℵ 0 . From Theorem 1.1 and by using standard technique (see for instance [4, Chapter VI, Section 12]) we get the following proposition in which n or m can be replaced by ℵ 0 : Proposition 1.2. Let G be a left R-module. Then the following assertions hold:

(1) there exists a (n, m)-pure exact sequence of left modules

0 → K → F → G → 0
where F is a direct sum of (m, n)-presented left modules;

(2) G is (n, m)-pure projective if and only if it is a summand of a direct sum of (m, n)-presented left modules.

A left R-module G is called (n, m)-pure-injective if for each (n, m)-pure exact sequence 0 → A → B → C → 0 the sequence 0 → Hom R (C, G) → Hom R (B, G) → Hom R (A, G) → 0 is exact. If M is a left module we put M ♯ = Hom Z (M, Q/Z). Thus M ♯ is a right module. It is the character module of M .
If A is a submodule of a left R-module B, we say that B is a (n, m)-pure essential extension of A if A is a (n, m)-pure submodule of B and for each nonzero submodule K of B such that A ∩ K = 0, (A + K)/K is not a (n, m)-pure submodule of B/K. If, in addition, B is (n, m)-pure injective, we say that B is a (n, m)-pure injective hull of A. In these above definitions and in the following proposition n or m can be replaced by ℵ 0 .

Proposition 1.3. The following assertions hold:

(1) each left R-module is a (n, m)-pure submodule of a (n, m)-pure injective left module;

(2) each left R-module has a (n, m)-pure injective hull which is unique up to an isomorphism.

Proof. [START_REF] Anderson | Coherence of power series rings over pseudo-Bezout domains[END_REF]. Let M be a left R-module. By Proposition 1.2 there exists a (m, n)pure exact sequence of right R-modules 0 → K → F → M ♯ → 0 where F is a direct sum of (n, m)-presented right modules. From Theorem 1.1 it follows that (M ♯ ) ♯ is a (n, m)-pure submodule of F ♯ . By [3, Corollary 1.30] M is isomorphic to a pure submodule of (M ♯ ) ♯ . So, M is isomorphic to a (n, m)-pure submodule of F ♯ . By using the canonical isomorphism (F ⊗ R -) ♯ ∼ = Hom R (-, F ♯ ) we get that F ♯ is (n, m)-pure injective since F is a direct sum of (n, m)-presented modules.

(2). Since (1) holds and every direct limit of (n, m)-pure exact sequences is (n, m)-pure exact too, we can adapt the method of Warfield's proof of existence of pure-injective hull to show (2)(see [START_REF] Warfield | Purity and algebraic compactness for modules[END_REF]Proposition 6]). We can also use [START_REF] Stenström | Pure submodules[END_REF]Proposition 4.5].

Proposition 1.4. Let R be a commutative ring and let (Σ) be a short exact sequence of R-modules. Then (Σ) is (n, m)-pure if and only if, for each maximal ideal P (Σ) P is (n, m)-pure.

Proof. Assume that (Σ) is (n, m)-pure and let M be a (n, m)-presented R P -module where P is a maximal ideal. There exists a (n, m)-presented R-module M ′ such that M ∼ = M ′ P and M ⊗ RP (Σ) P ∼ = (M ′ ⊗ R (Σ)) P . We deduce that (Σ) P is (n, m)-pure. Conversely, suppose that (Σ) is the sequence 0 → A → B → C → 0. Let M be a (n, m)-presented R-module. Then, for each maximal ideal P , (Σ) P is (n, m)-pure over R since M ⊗ R (Σ) P ∼ = M P ⊗ RP (Σ) P . On the other hand, since M ⊗ R ( P ∈Max R (Σ) P ) ∼ = ( P ∈Max R M ⊗ R (Σ) P ), P ∈Max R A P is a (n, m)-pure submodule of P ∈Max R B P . By [2, Lemme 1.3] A is isomorphic to a pure submodule of P ∈Max R A P . We successively deduce that A is a (n, m)-pure submodule of P ∈Max R B P and B.

Comparison of purities over a semiperfect ring

In this section we shall compare (n, m)-purities for different pairs of integers (n, m). In [START_REF] Puninski | Rings described by various purities[END_REF] some various purities are also compared. In particular some necessary and sufficient conditions on a ring R are given for the (1, 1)-purity to be equivalent to the (ℵ 0 , ℵ 0 )-purity.

The following lemma is due to Lawrence Levy, see [START_REF] Wiegand | Finitely generated modules over Bezout rings[END_REF]Lemma 1.3]. If M be a finitely generated left (or right) R-module, we denote by gen M its minimal number of generators.

Lemma 2.1. Let R be a ring. Assume there exists a positive integer p such that gen A ≤ p for each finitely generated left ideal A of R. Then gen N ≤ p × gen M , if N is a finitely generated submodule of a finitely generated left R-module M .

From this lemma and Theorem 1.1 we deduce the following: Proposition 2.2. Let R be a ring. Assume there exists a positive integer p such that gen A ≤ p for each finitely generated left ideal A of R. Then, for each positive integer n:

(1) each (n, np)-pure exact sequence of right modules is (n, ℵ 0 )-pure exact;

(2) each (np, n)-pure exact sequence of left modules is (ℵ 0 , n)-pure exact.

Corollary 2.3. Let R be a left Artinian ring. Then there exists a positive integer p such that, for each positive integer n:

(1) each (n, np)-pure exact sequence of right modules is (n, ℵ 0 )-pure exact;

(2) each (np, n)-pure exact sequence of left modules is (ℵ 0 , n)-pure exact.

Proof. Each finitely generated left R-module M has a finite length denoted by length M , and gen M ≤ length M . So, for each left ideal A we have gen A ≤ length R. We choose p = sup{gen A | A left ideal of R} and we apply the previous proposition.

Let R be a ring and J its Jacobson radical. Recall that R is semiperfect if R/J is semisimple and idempotents lift modulo J.

Theorem 2.4. Let R be semiperfect ring. Assume that each indecomposable finitely presented cyclic left R-module has a local endomorphism ring. The following conditions are equivalent:

(1) there exists an integer p > 0 such that, for each integer n > 0, each (n, np)pure exact sequence of right modules is (n, ℵ 0 )-pure exact; (2) there exists an integer p > 0 such that, for each integer n > 0, each (np, n)pure exact sequence of left modules is (ℵ 0 , n)-pure exact; (3) there exists an integer q > 0 such that each (1, q)-pure exact sequence of right modules is (1, ℵ 0 )-pure exact; (4) there exists an integer q > 0 such that each (q, 1)-pure exact sequence of left modules is (ℵ 0 , 1)-pure exact; (5) there exists an integer q > 0 such that each indecomposable finitely presented cyclic left module is q-related;

(6) there exists an integer p > 0 such that gen A ≤ p for each finitely generated left ideal A of R. Moreover, if each indecomposable finitely presented left R-module has a local endomorphism ring, these conditions are equivalent to the following: [START_REF] Puninski | Rings described by various purities[END_REF] there exist two positive integers n, m such that each (n, m)-pure exact sequence of right modules is (n, ℵ 0 )-pure exact; (8) there exist two positive integers n, m such that each (m, n)-pure exact sequence of left modules is (ℵ 0 , n)-pure exact;

Proof. By Proposition 2.2 ( 6) ⇒ (1). By Theorem 1.1 (1) ⇔ (2), ( 3) ⇔ ( 4) and ( 7) ⇔ (8). It is obvious that (2) ⇒ (4) and ( 2) ⇒ (7). ( 4) ⇒ (5). Let C be an indecomposable finitely presented cyclic left module. Then C is (q, 1)-pure-projective. So, C is a direct summand of a finite direct sum of (1, q)-presented left modules. Since R is semiperfect, we may assume that these (1, q)-presented left modules are indecomposable. So, by Krull-Schmidt theorem C is (1, q)-presented.

(5) ⇒ (6). Let A be a finitely generated left ideal. Then R/A = ⊕ t i=1 R/A i where, for each i = 1, . . . , t, A i is a left ideal and R/A i is indecomposable. We have the following commutative diagram with exact horizontal sequences:

0 → ⊕ t i=1 A i → R t → ⊕ t i=1 R/A i → 0 ↓ ↓ ↓ 0 → A → R → R/A → 0
Since the right vertical map is an isomorphism, we deduce from the snake lemma that the other two vertical homomorphisms have isomorphic cokernels. It follows that gen A ≤ tq + 1 because gen A i ≤ q by (5). On the other hand, let P be a projective cover of R/A. Then P is isomorphic to a direct summand of R. We know that the left module R is a finite direct sum of indecomposable projective modules. Let s the number of these indecomposable summands. It is easy to show that t ≤ s. So, if p = sq + 1, then gen A ≤ p. (8) ⇒ (5). Let C be an indecomposable finitely presented cyclic left module. Then C is (m, n)-pure-projective. So, C is a direct summand of a finite direct sum of (n, m)-presented left modules. Since R is semiperfect, we may assume that these (n, m)-presented left modules are indecomposable. So, by the Krull-Schmidt theorem C is (1, m)-presented.

A ring R is said to be strongly π-regular if, for each r ∈ R, there exist s ∈ R and an integer q ≥ 1 such that r q = r q+1 s. By [START_REF] Facchini | Module theory. Endomorphism rings and direct sum decompositions in some classes of modules[END_REF]Theorem 3.16] each strongly π-regular R satisfies the following condition: for each r ∈ R, there exist s ∈ R and an integer q ≥ 1 such that r q = sr q+1 . Recall that a left R-module M is said to be Fitting if for each endomorphism f of M there exists a positive integer t such that M = ker f t ⊕ f t (M ). Lemma 2.5. Let R be a strongly π-regular semiperfect ring. Then:

(1) each finitely presented cyclic left (or right) R-module is Fitting;

(2) each indecomposable finitely presented cyclic left (or right) R-module has a local endomorphism ring.

Proof. In [3, Lemma 3.21] it is proven that every finitely presented R-module is a Fitting module if R is a semiperfect ring with M n (R) strongly π-regular for all n.

We do a similar proof to show (1).

(2). By [3, Lemma 2.21] each indecomposable Fitting module has a local endomorphism ring.

Corollary 2.6. Let R be a strongly π-regular semiperfect ring. The following conditions are equivalent:

(1) there exists an integer p > 0 such that, for each integer n > 0, each (n, np)pure exact sequence of right modules is (n, ℵ 0 )-pure exact; (2) there exists an integer p > 0 such that, for each integer n > 0, each (np, n)pure exact sequence of left modules is (ℵ 0 , n)-pure exact;

(3) there exists an integer q > 0 such that each (1, q)-pure exact sequence of right modules is (1, ℵ 0 )-pure exact; (4) there exists an integer q > 0 such that each (q, 1)-pure exact sequence of left modules is (ℵ 0 , 1)-pure exact; (5) there exists an integer q > 0 such that each indecomposable finitely presented cyclic left module is q-related; (6) there exists an integer p > 0 such that gen A ≤ p for each finitely generated left ideal A of R.

Moreover, if M n (R) is strongly π-regular for all n > 0, these conditions are equivalent to the following: Recall that a ring R is right perfect if each flat right R-module is projective.

Corollary 2.7. Let R be a right perfect ring. The following conditions are equivalent:

(1) there exists an integer p > 0 such that, for each integer n > 0, each (n, np)pure exact sequence of right modules is (n, ℵ 0 )-pure exact; (2) there exists an integer p > 0 such that, for each integer n > 0, each (np, n)pure exact sequence of left modules is (ℵ 0 , n)-pure exact; (3) there exists an integer q > 0 such that each (1, q)-pure exact sequence of right modules is (1, ℵ 0 )-pure exact; (4) there exist two positive integers n, m such that each (n, m)-pure exact sequence of right modules is (n, ℵ 0 )-pure exact; (5) there exist two positive integers n, m such that each (m, n)-pure exact sequence of left modules is (ℵ 0 , n)-pure exact; (6) there exists an integer p > 0 such that gen A ≤ p for each finitely generated left ideal A of R.

Proof. For all n > 0, M n (R) is right perfect. Since each right perfect ring satisfies the descending chain condition on finitely generated left ideals, then M n (R) is strongly π-regular for all n > 0. We apply Corollary 2.6.

Comparison of purities over a commutative ring

In the sequel of this section R is a commutative local ring, except in Theorem 3.5. We denote respectively by P and k its maximal ideal and its residue field Let M be a finitely presented R-module. Recall that gen M = dim k M/P M . Let F 0 be a free R-module whose rank is gen M and let φ : F 0 → M be an epimorphism. Then ker φ ⊆ P F 0 . We put rel M = gen ker φ. Let F 1 be a free R-module whose rank is rel M and let f : F 1 → F 0 be a homomorphism such that im f = ker φ. Then ker f ⊆ P F 1 . For any R-module N , we put

N * = Hom R (N, R). Let f * : F * 0 → F * 1 be
(4) if M = M 1 ⊕ M 2 then gen M = gen M 1 + gen M 2 and rel M = rel M 1 + rel M 2 . (5) End R (D(M )) is local if and only if so is End R (M ).
Lemma 3.2. Let M be a finitely generated R-module, s an endomorphism of M and s the endomorphism of M/P M induced by s. Then s is an isomorphism if and only if so is s.

Proof. If s is an isomorphism it is obvious that so is s. Conversely, coker s = 0 by Nakayama lemma. So, s is surjective. By using a Vasconcelos's result (see [4, Theorem V.2.3]) s is bijective. Proposition 3.3. Assume that there exists an ideal A with gen A = p + 1 where p is a positive integer. Then, for each positive integers n and m with (n -1)p + 1 ≤ m ≤ np+ 1, there exists a finitely presented R-module W p,n,m whose endomorphism ring is local and such that gen W p,n,m = n and rel W p,n,m = m.

Proof. Suppose that A is generated by a 1 , . . . , a p , a p+1 . Let F be a free module of rank n with basis e 1 , . . . , e n and let K be the submodule of F generated by x 1 , . . . , x m where these elements are defined in the following way: if j = pq + r where 1 ≤ r ≤ p, x j = a r e q+1 if r = 1 or q = 0, and x j = a p+1 e q + a 1 e q+1 else; when m = pn + 1, x m = a p+1 e n . We put W p,n,m = F/K. We can say that W p,n,m is named by the following n × m matrix, where r = m -p(n -1): Since {e 1 , . . . , e n } is a basis and gen A = p+1 we deduce that c j ∈ P, ∀j, 1 ≤ j ≤ m. So, rel W p,n,m = m. Let s ∈ End R (W p,n,m ). Then s is induced by an endomorphism s of F which satisfies s(K) ⊆ K. For each j, 1 ≤ j ≤ n, there exists a family (α i,j ) of elements of R such that: [START_REF] Anderson | Coherence of power series rings over pseudo-Bezout domains[END_REF] s(e j ) =

      
n i=1 α i,j e i Since s(K) ⊆ K, ∀j, 1 ≤ j ≤ m, ∃ a family (β i,j ) of elements of R such that: (2) s(x j ) = m i=1 β i,j x i
From ( 1), ( 2) and the equality x 1 = a 1 e 1 if follows that:

n q=1 α q,1 a 1 e q = n-2 q=0 p+1 i=1 β pq+i,1 a i e q+1 + r i=1 β p(n-1)+i,1 a i e n .
Then, we get:

∀q, 1 ≤ q ≤ n -1, α q,1 a 1 = p+1 i=1 β p(q-1)+i,1 a i and α n,1 a 1 = r i=1 β p(n-1)+i,1 a i .
We deduce that: ∀q, 2 ≤ q ≤ n, β p(q-2)+p+1,1 ∈ P and β p(q-1)+1,1 ≡ α q,1 [P ]. So,

(3) ∀q, 2 ≤ q ≤ n, α q,1 ∈ P. Now, let j = pℓ + 1 where 1 ≤ ℓ ≤ (n -1). In this case, x j = a p+1 e ℓ + a 1 e ℓ+1 . From ( 1) and ( 2) it follows that: n q=1 (α q,ℓ a p+1 + α q,ℓ+1 a 1 )e q = n-2 q=0 p+1 i=1 β pq+i,j a i e q+1 + r i=1 β p(n-1)+i,j a i e n .

Then, we get:

∀q, 1 ≤ q ≤ n -1, α q,ℓ a p+1 + α q,ℓ+1 a 1 = p+1 i=1 β p(q-1)+i,j a i and α n,ℓ a p+1 + α n,ℓ+1 a 1 = r i=1 β p(n-1)+i,j a i .
We deduce that ∀q, ℓ, 1 ≤ q, ℓ ≤ (n -1), α q,ℓ ≡ β p(q-1)+p+1,j [P ] and α q+1,ℓ+1 ≡ β pq+1,j [P ],

whence α q,ℓ ≡ α q+1,ℓ+1 [P ]. Consequently, ∀q, 1 ≤ q ≤ n, α q,q ≡ α 1,1 [P ] and ∀t, 1 ≤ t ≤ (n -1), ∀q, 1 ≤ q ≤ (n -t), α q+t,q ≡ α 1+t,1 ≡ 0 [P ] by [START_REF] Facchini | Module theory. Endomorphism rings and direct sum decompositions in some classes of modules[END_REF] Theorem 3.5. Let R be a commutative ring. The following assertions hold:

(1) Assume that, for any integer p > 0, there exists a maximal ideal P and a finitely generated ideal A of R P such that gen RP A ≥ p + 1. Then, if (n, m) and (r, s) are two different pairs of integers, the (n, m)-purity and the (r, s)-purity are not equivalent.

(2) Assume that, there exists an integer p > 0 such that, for each maximal ideal P , for any finitely generated ideal A of R P , gen RP A ≤ p. Then: (a) for each integer n > 0 the (ℵ 0 , n)-purity (respectively (n, ℵ 0 )-purity) is equivalent to the (np, n)-purity (respectively (n, np)-purity); (b) if p > 1, then, for each integer n > 0, for each integer m, 1 ≤ m ≤ n(p-1), the (n, m)-purity (respectively (m, n)-purity) is not equivalent to the (n, m + 1)-purity (respectively (m + 1, n)-purity).

Proof. By Proposition 1.4 we may assume that R is local with maximal P . By Theorem 1.1 the (n, m)-purity and the (r, s)-purity are equivalent if and only if so are the (m, n)-purity and the (s, r)-purity.

(1). Suppose that r > n and let t = min(m, s). Let q be the greatest divisor of (r -1) which is ≤ t and p = (r -1)/q. Let A be a finitely generated ideal such that gen A > p. By way of contradiction, suppose that W p,q,r is (n, m)pure-projective. By Proposition 1.2 W p,q,r is a summand of ⊕ i∈I F i where I is a finite set and ∀i ∈ I, F i is a (m, n)-presented R-module. Since its endomorphism ring is local, W p,q,r is an exchange module (see [START_REF] Facchini | Module theory. Endomorphism rings and direct sum decompositions in some classes of modules[END_REF]Theorem 2.8]). So, we have

W p,q,r ⊕ (⊕ i∈I G i ) ∼ = (⊕ i∈I H i ) ⊕ (⊕ i∈I G i ) where ∀i ∈ I, G i and H i are submodules of F i and F i = G i ⊕ H i . Let G = ⊕ i∈I G i .
Then G is finitely generated. By [4, Proposition V.7.1] End R (G) is semilocal. By using Evans's theorem ([3, Corollary 4.6]) we deduce that W p,q,r ∼ = (⊕ i∈I H i ). Since W p,q,r is indecomposable, we get that it is (m, n)-presented. This contradicts that rel W p,q,r = r > n.

(2)(a) is an immediate consequence of Proposition 2.2.

(2)(b). There exist two integers q, t such that m + 1 = (q -1)(p -1) + t with n ≥ q ≥ 1 and 1 ≤ t ≤ p. As in [START_REF] Anderson | Coherence of power series rings over pseudo-Bezout domains[END_REF] we prove that W p-1,q,m+1 is not (m, n)-pureprojective.

Remark 3.6. In the previous theorem, when there exists an integer p > 1 such that, for any finitely generated ideal A gen A ≤ p, we don't know if the (n, m)-purity and the (n, m + 1)-purity are equivalent when n(p -1) + 1 ≤ m ≤ np -1. If R is a local ring with maximal P with residue field k such that P 2 = 0 and dim k P = p it is easy to show that each finitely presented R-module F with gen F = n and rel F = np is semisimple. So, the (np, n)-purity is equivalent to the (np -1, n)-purity.

(n, m)-flat modules and (n, m)-injective modules

Let M be a right R-module. We say that M is (n, m)-flat if for any m-generated submodule K of a n-generated free left R-module F , the natural map:

M ⊗ R K → M ⊗ R F is a monomorphism. We say that M is (ℵ 0 , m)-flat (respectively (n, ℵ 0 )- flat) if M is (n, m)-flat for each integer n > 0 (respectively m > 0). We say that M is (n, m)-injective if for any m-generated submodule K of a n-generated free right R-module F , the natural map: Hom R (F, M ) → Hom R (K, M ) is an epimorphism. H = G ∩ R n .
We have the following commutative diagram with exact horizontal sequences:

M ⊗ R H → M ⊗ R G 1M ⊗π ----→ M ⊗ R G ′ → 0 ↓ ↓ ↓ 0 → M ⊗ R R n → M ⊗ R R n+1 1M ⊗π ----→ M ⊗ R R → 0 Let u : G → R n+1 , u ′ : G ′ → R, w : R n → R n+1 be the inclusion maps and let v = u| H . Then (1 M ⊗ u ′ ) is injective. Let H ′ be a p-generated submodule of H. By the induction hypothesis M is (n, p)-flat. So, (1 M ⊗ (v| H ′ )) is injective. It follows that (1 M ⊗ v) is injective too. We conclude that (1 M ⊗ u) is injective and M is (ℵ 0 , p)-flat.
Corollary 4.6. Let p be a positive integer and let R be a ring such that each left ideal is (1, p)-flat. Then, for each positive integer q ≤ p, a right R-module M is (1, q)-flat if and only if it is (ℵ 0 , q)-flat.

Proof. Let the notations be as in the previous theorem. Since G ′ is a flat left Rmodule by Lemma 4.4, H is a pure submodule of G. Let {g 1 , . . . , g q } be a spanning set of G and let h 1 , . . . , h t ∈ H. For each k, 1 ≤ k ≤ t there exist a k,1 , . . . , a k,q ∈ R such that h k = q i=1 a k,i g i . It follows that there exist g ′ 1 , . . . , g ′ q ∈ H such that ∀k,

1 ≤ k ≤ t, h k = q i=1 a k,i g ′ i .
So, each finitely generated submodule of H is contained in a q-generated submodule. We conclude by applying Theorem 4.5.

Corollary 4.7. Let R be a commutative local ring with maximal P . Assume that P 2 = 0. Let q a positive integer. Then:

(1) each (1, q)-flat module is (ℵ 0 , q)-flat;

(2) each (1, q)-injective module is (ℵ 0 , q)-injective.

Proof. Let the notations be as in the previous theorem. We may assume that G ⊆ P R n+1 . Then G is a semisimple module and H is a direct summand of G. So, (1) is a consequence of Theorem 4.5.

(2). Let M be a (1, q)-injective module. We shall prove by induction on n that M is (n, q)-injective. We have the following commutative diagram:

0 → Hom R (R, M ) → Hom R (R n+1 , M ) → Hom R (R n , M ) → 0 ↓ ↓ ↓ 0 → Hom R (G ′ , M ) → Hom R (G, M ) → Hom R (H, M ) → 0
where the horizontal sequences are exact. By the induction hypothesis the left and the right vertical maps are surjective. It follows that the middle vertical map is surjective too.

By [START_REF] Shamsuddin | n-injective and n-flat modules[END_REF]Example 5.2] or [6, Theorem 2.3], for each integer n > 0, there exists a ring R for which each finitely generated left ideal is (1, n)-flat (hence (ℵ 0 , n)-flat by Corollary 4.6) but there is a finitely generated left ideal which is not (1, n + 1)-flat. The following proposition gives other examples in the commutative case. Proposition 4.8. Let R be a commutative local ring with maximal ideal P and residue field k. Assume that P 2 = 0 and dim k P > 1. Then, for each positive integer p < dim k P , there exists:

(1) a (p + 1, 1)-presented R-module which is (ℵ 0 , p)-flat but not (1, p + 1)-flat;

(2) a (ℵ 0 , p)-injective R-module which is not (1, p + 1)-injective.

Proof. [START_REF] Anderson | Coherence of power series rings over pseudo-Bezout domains[END_REF]. Let F be a free R-module of rank (p + 1) with basis {e 1 , . . . , e p , e p+1 }, let (a 1 , . . . , a p , a p+1 ) be a family of linearly independent elements of P , let K be the submodule of F generated by p+1 i=1 a i e i and let M = F/K. Then M ∼ = D(W p,1,p+1 ) (see the proof of Proposition 3.3). First, we show that K is a (1, p)-pure submodule of F . We consider the following equation:

(4) p j=1 r j x j = s( p+1 i=1 a i e i )
where r 1 , . . . , r p , s ∈ R and with unknowns x 1 , . . . , x p . Assume that this equation has a solution in F . Suppose there exists ℓ, 1 ≤ ℓ ≤ p, such that r ℓ is a unit. For each j, 1 ≤ j ≤ p, we put x ′ j = δ j,ℓ r -1 ℓ s( p+1 i=1 a i e i ). It is easy to check that (x ′ 1 , . . . , x ′ p ) is a solution of (4) in K. Now we assume that r j ∈ P, ∀j, 1 ≤ j ≤ p. Suppose that (x 1 , . . . , x p ) is a solution of (4) in F . For each j, 1 ≤ j ≤ p, x j = p+1 i=1 c j,i e i , where c j,i ∈ R. We get the following equality:

(5)

p+1 i=1   p j=1 r j c j,i   e i = p+1 i=1 sa i e i
We deduce that:

(6) ∀i, 1 ≤ i ≤ p + 1, p j=1 r j c j,i = sa i So, if s is a unit, ∀i, 1 ≤ i ≤ p + 1, a i ∈ p j=1 Rr j . It follows that dim k p+1 i=1
Ra i ≤ p that is false. So, s ∈ P . In this case (4) has the nil solution. Hence M is (ℵ 0 , p)-flat by Proposition 4.1(2) and Corollary 4.7. By way of contradiction suppose that M is (1, p + 1)-flat. It follows that K is a (1, p + 1)-pure submodule of F by Proposition 4.1. Since M is (1, p + 1)-pureprojective we deduce that M is free. This is false.

(2). Let E be an injective R-cogenerator. Then Hom R (M, E) is (ℵ 0 , p)-injective but not (1, p + 1)-injective by Proposition 4.1(4).

In a similar way we show the following proposition. Proposition 4.9. Let R be a commutative local ring with maximal ideal P . Assume that P 2 = 0. Let M be a (m, 1)-presented R-module with m > 1, let {x 1 , . . . , x m } be a spanning set of M and let m j=1 a j x j = 0 be the relation of M , where a 1 , . . . , a m ∈ P

. If p = gen ( m j=1 Ra j ) -1 > 0, then: (1) M is (ℵ 0 , p)-flat but not (1, m)-flat; (2) Hom R (M, E) is (ℵ 0 , p)-injective but not (1, m)-injective, where E is an injective R-cogenerator.
When R is an arithmetical commutative ring, i.e. its lattice of ideals is distributive, each (1, 1)-flat module is flat and by [4, Theorem VI.9.10] the converse holds if R is a commutative domain (it is also true if each principal ideal is flat). However we shall see that there exist non-arithmetical commutative rings for which each (1, 1)-flat module is flat. Recall that a left (or right) R-module M is torsionless if the natural map M → (M * ) * is injective. Proposition 4.10. For each ring R the following conditions are equivalent:

(1) R is right self (ℵ 0 , 1)-injective;

(2) each finitely presented cyclic left R-module is torsionless;

(3) each finitely generated left ideal A satisfies A = l -ann(r -ann(A)).

Proof. We prove (1) ⇔ (2) as [5, Theorem 2.3] and (2) ⇔ (3) is easy.

Theorem 4.11. Let R be a right perfect ring which is right self (ℵ 0 , 1)-injective.

Then each (1, 1)-flat right module is projective.

Proof. Let M be a (1, 1)-flat right R-module. It is enough to show that M is flat. Let A be a finitely generated left ideal of R. Assume that {a 1 , . . . , a n } is a minimal system of generators of A with n > 1. Let z ∈ M ⊗ R A such that its image in M is 0. We have z = n i=1 y i ⊗ a i , where y 1 , . . . , y n ∈ M , and n i=1 y i a i = 0. For each i, 1 ≤ i ≤ n, we set A i = n ( j=1 j =i ) Ra j . Then, ∀i, 1 ≤ i ≤ n, A i ⊂ A. For each finitely generated left ideal B we have B = l -ann(r -ann(B)). It follows that, ∀i, 1 ≤ i ≤ n, r -ann(A) ⊂ r -ann(A i ). Let b i ∈ r -ann(A i )) \ r -ann(A). Then y i a i b i = 0. From the (1, 1)-flatness of M we deduce that y i = mi k=1 y ′ i,k c i,k , where y ′ i,1 , . . . , y ′ i,mi ∈ M and c i,1 , . . . , c i,mi ∈ R with c i,k a i b i = 0, ∀k, 1 ≤ k ≤ m i . It follows that z = n i=1 ( mi k=1 y ′ i,k ⊗ c i,k a i ). Let A (1] be the left ideal generated by {c i,k a i | 1 ≤ i ≤ n, 1 ≤ k ≤ m i }. Then A (1) ⊂ A; else, ∀i, 1 ≤ i ≤ n, a i = n j=1 ( mj k=1 d i,j,k c j,k a j ) with d i,j,k ∈ R; we get that a i b i = n j=1 ( mj k=1 d i,j,k c j,k a j b i ); but a j b i = 0 if j = i and c i,k a i b i = 0; so, there is a contradiction because the second member of the previous equality is 0 while a i b i = 0 . Let {a

n1 } be a minimal system of generators of A (1) . So, z = n1 i=1 y

(1) i ⊗a [START_REF] Anderson | Coherence of power series rings over pseudo-Bezout domains[END_REF] i where y

n1 ∈ M , and z is the image of z (1) ∈ M ⊗ R A (1) defined by z (1) = n1 i=1 y

(1) i ⊗ a

(1)

i . If n 1 ≤ 1 we conclude that z (1) = 0 since M is (1, 1)-flat, and z = 0. If n 1 > 1, in the same way we get that z (1) is the image of an element z (2) ∈ M ⊗ R A (2) where A (2) is a left ideal such that A (2) ⊂ A (1) . If gen A (2) > 1 we repeat this process, possibly several times, until we get a left ideal A (l) with gen A (l) ≤ 1; this is possible because R satisfies the descending chain condition on finitely generated left ideals since it is right perfect (see [START_REF] Renault | Algèbre non commutative. Collection "Varia Mathematica[END_REF]Théorème 5 p.130]). The (1, 1)-flatness of M implies that z (l) = 0 and z = 0. So, M is projective.

Let P be a ring property. We say that a commutative ring R is locally P if R P satisfies P for each maximal ideal P .

The following corollary is a consequence of Theorem 4.11 and Proposition 4.3.

Corollary 4.12. Let R be a commutative ring which is locally perfect and locally self (ℵ 0 , 1)-injective. Then each (1, 1)-flat R-module is flat.

(n, m)-coherent rings

We say that a ring R is left (n, m)-coherent if each m-generated submodule of a n-generated free left R-module is finitely presented. We say that R is left (ℵ 0 , m)coherent (respectively (n, ℵ 0 )-coherent) if for each integer n > 0 (respectively m > 0) R is left (n, m)-coherent. The following theorem can be proven with standard

( 7 )

 7 there exist two positive integers n, m such that each (n, m)-pure exact sequence of right modules is (n, ℵ 0 )-pure exact; (8) there exist two positive integers n, m such that each (m, n)-pure exact sequence of left modules is (ℵ 0 , n)-pure exact; Proof. By Lemma 2.5 each indecomposable finitely presented cyclic left R-module has a local endomorphism ring. If M n (R) is strongly π-regular for all n, then by [3, Lemmas 3.21 and 2.21] each indecomposable finitely presented left R-module has a local endomorphism ring. So, we apply Theorem 2.4.

Proposition 3 . 1 .

 31 the homomorphism deduced from f . We set D(M ) = coker f * the Auslander and Bridger's dual of M . The following proposition is version in commutative case of [14, Theorem 2.4]: Assume that M has no projective summand. Then: (1) ker f * ⊆ P F * 0 and im f * ⊆ P F * 1 ; (2) M ∼ = D(D(M )) and D(M ) has no projective summand; (3) gen D(M ) = rel M and rel D(M ) = gen M ;

a 1 .a 1 a 2 .c

 12 . a p a p+1 0 . . . . . . . . . . . . . . . . . . . . 0 .. 0 a 1 a 2 . . . a p+1 0 .. . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 a 1 .. a p a p+1 0 .. 0 . . . . . . . . . . 0 Since K ⊆ P F , gen W p,n,m = n. Now we consider the following relation: m j=0 c j x j = 0. From the definition of the x j we get the following equality: pq+i a i e q+1 + r i=1 c p(n-1)+i a i e n = 0.

  . Let s be the endomorphism of W p,n,m /P W p,n,m induced by s. If α 1,1 is a unit then s is an isomorphism, else 1 Wp,n,m -s is an isomorphism. By Lemma 3.2 we conclude that either s or (1 Wp,n,m -s) is an isomorphism. Hence, End R (W p,n,m ) is local. Remark 3.4. Observe that D(W 1,n-1,n ) is isomorphic to the indecomposable module built in the proof of[START_REF] Warfield | Decomposability of finitely presented modules[END_REF] Theorem 2].

We say that M is (ℵ 0 , m)-injective (respectively (n, ℵ 0 )-injective) if M is (n, m)injective for each integer n > 0 (respectively m > 0). A ring R is called left self (n, m)-injective if R is (n, m)-injective as left R-module.

If R is a commutative domain, then an R-module is (1, 1)-flat (respectively (1, 1)injective) if and only if it is torsion-free (respectively divisible).

The following propositions can be proved with standard technique: see [16, Theorem 4.3 and Proposition 2.3]. In these propositions the integers n or m can be replaced with ℵ 0 . Proposition 4.1. Assume that R is an algebra over a commutative ring S and let E be an injective S-cogenerator. Let M be a right R-module. The following conditions are equivalent:

(1) M is (n, m)-flat;

(2) ( Proof. Only "if" requires a proof. Let A be a left ideal. Assume that M is generated by

Since M is (1, p)-flat we successively deduce that z ′ = 0 and z = 0.

It is well known that each (1, ℵ 0 )-flat right module is (ℵ 0 , ℵ 0 )-flat. For each positive integer p, is each (1, p)-flat right module (ℵ 0 , p)-flat?

The following theorem and Theorem 4.11 give a partial answer to this question.

Theorem 4.5. Let p be a positive integer and let R be a ring. For each positive integer n, assume that, for each p-generated submodule

Proof. We shall prove that M is (n, p)-flat by induction on n. Let G be a p-

Then G ′ is a p-generated left module. We put technique: see [START_REF] Zhang | On (m, n)-injective modules and (m, n)-coherent rings[END_REF]Theorems 5.1 and 5.7]. In this theorem the integers n or m can be replaced with ℵ 0 .

Theorem 5.1. Let R be a ring and n, m two fixed positive integers. Assume that R is an algebra over a commutative ring S. Let E be an injective S-cogenerator.

Then the following conditions are equivalent:

(1) R is left (n, m)-coherent;

(2) any direct product of right (n, m)-flat R-modules is (n, m)-flat;

(3) for any set Λ, R Λ is a (n, m)-flat right R-module; (4) any direct limit of a direct system of (n, m)-injective left R-modules is (n, m)-injective; (5) for any exact sequence of left modules 0

For each positive integer p, is each left (1, p)-coherent ring left (ℵ 0 , p)coherent?

Propositions 5.2 and 5.3 and Theorem 5. [START_REF] Fuchs | Modules over Non-Noetherian Domains[END_REF] give a partial answer to this question.

Proposition 5.2. Let p be a positive integer and let R be a ring. For each positive integer n, assume that, for each p-generated submodule G of the left R-module R n ⊕ R, (G ∩ R n ) is the direct limit of its p-generated submodules. Then the following conditions are equivalent:

(1) R is left (1, p)-coherent;

(2) R is left (ℵ 0 , p)-coherent. Moreover, when these conditions hold each (1, p)-injective left module is (ℵ 0 , p)injective.

Proof. It is obvious that (2) ⇒ (1).

(1) ⇒ (2). Let Λ be a set. By Theorem 5.1 R Λ is a (1, p)-flat right module. From Theorem 4.5 we deduce that R Λ is a (ℵ 0 , p)-flat right module. By using again Theorem 5.1 we get [START_REF] Couchot | Exemples d'anneaux auto fp-injectifs[END_REF].

Let M be a (1, p)-injective left module. By Theorem 5.1 M ♯ is a (1, p)-flat right R-module. Then it is also (ℵ 0 , p)-flat. We deduce that (M ♯ ) ♯ is a (ℵ 0 , p)-injective left module. Since M is a pure submodule of (M ♯ ) ♯ , it follows that M is (ℵ 0 , p)injective too.

Proposition 5.3. Let R be a commutative perfect ring. Then R is Artinian if and only if it is (1, 1)-coherent.

Proof. Suppose that R is (1, 1)-coherent. Since R is perfect, R is a finite product of local rings. So, we may assume that R is local with maximal P . Let S be a minimal non-zero ideal of R generated by s. Then P is the annihilator of s. So, P is finitely generated and it is the sole prime ideal of R. Since all prime ideals of R are finitely generated, R is Noetherian. On the other hand R satisfies the descending chain condition on finitely generated ideals. We conclude that R is Artinian.

Except in some particular cases, we don't know if each (1, p)-injective module is (ℵ 0 , p)-injective, even if we replace p by ℵ 0 .

Theorem 5.4. Let R be a ring which is right perfect, left (1, 1)-coherent and right self (ℵ 0 , 1)-injective. Then each (1, 1)-injective left module is (ℵ 0 , ℵ 0 )-injective and R is left coherent.

Proof. Let M be a left (1, 1)-injective module. By Theorem 5.1 M ♯ is (1, 1)-flat. Whence M ♯ is projective by Theorem 4.11. We do as in the proof of Proposition 5.2 to conclude that M is (ℵ 0 , ℵ 0 )-injective.

For each set Λ, R Λ is a (1, 1)-flat right module by Theorem 5.1. It follows that R Λ is a projective right module by Theorem 4.11.

Recall that a ring is quasi-Frobenius if it is Artinian and self-injective.

Corollary 5.5. Let R be a quasi-Frobenius ring. Then, for each right (or left) R-module M , the following conditions are equivalent:

(1) M is (1, 1)-flat;

(2) M is projective;

(3) M is injective;

Proof. It is well known that (2) ⇔ (3). By Theorem 4.11 (1) ⇔ (2) because R satisfies the conditions of this theorem, and it is obvious that (3) ⇒ (4) and the converse holds by Theorem 5.4.

We prove the following theorem as [2, Théorème

Theorem 5.6. Let R be a commutative ring and n, m two fixed positive integers.

The following conditions are equivalent:

(1) R is (n, m)-coherent;

(2) for each multiplicative subset S of R, S -1 R is (n, m)-coherent, and for each (n, m)-injective R-module M , S -1 M is (n, m)-injective over S -1 R; (3) For each maximal ideal P , R P is (n, m)-coherent and for each (n, m)injective R-module M , M P is (n, m)-injective over R P .

Recall that a ring R is a right IF-ring if each right injective R-module is flat.

Theorem 5.7. Let R be a commutative ring which is locally perfect, (1, 1)-coherent and self (1, 1)-injective. Then:

(1) R is coherent, self (ℵ 0 , ℵ 0 )-injective and locally quasi-Frobenius;

(2) each (1, 1)-flat module is flat;

(3) each (1, 1)-injective module is (ℵ 0 , ℵ 0 )-injective.

Proof. By Theorem 5.6 R P is (1, 1)-coherent and (1, 1)-injective for each maximal ideal P . Let a be a generator of a minimal non-zero ideal of R P . Then P R P is the annihilator of a and consequently P R P is finitely generated over R P . Since all prime ideals of R P are finitely generated, we deduce that R P is Artinian for each maximal ideal P . Moreover, the (1, 1)-injectivity of R P implies that the socle of R P (the sum of all minimal non-zero ideals) is simple. It follows that R P is quasi-Frobenius for each maximal ideal P . Let M be a (ℵ 0 , ℵ 0 )-injective R-module. By Theorem 5.6 M P is (1, 1)-injective for each maximal ideal P . By Corollary 5.5 M P is injective for each maximal ideal P . We conclude that R is self (ℵ 0 , ℵ 0 )-injective and it is coherent by Theorem 5.6.

If M is (1, 1)-injective, we prove as above that M P is injective for each maximal ideal P . It follows that M is (ℵ 0 , ℵ 0 )-injective.

The second assertion is an immediate consequence of Corollary 4.12.

The following proposition is easy to prove: Proposition 5.8. A ring R is left (ℵ 0 , 1)-coherent if and only if each finitely generated right ideal has a finitely generated left annihilator.

Example 5.9. Let V be a non-Noetherian (commutative) valuation domain whose order group is not the additive group of real numbers and let R = V [[X]] be the power series ring in one indeterminate over V . Since R is a domain, R is (ℵ 0 , 1)coherent. But, in [START_REF] Anderson | Coherence of power series rings over pseudo-Bezout domains[END_REF] it is proven that there exist two elements f and g of R such that Rf ∩ Rg is not finitely generated. By using the exact sequence 0 → Rf ∩ Rg → Rf ⊕ Rg → Rf + Rg → 0 we get that Rf + Rg is not finitely presented. So, R is not (1, 2)-coherent.


Maxime Clusel
email: maxime.clusel@univ-montp2.fr

Eric I Corwin
email: ecorwin@uoregon.edu

Unfolding protein with an atomic force microscope: Force-fluctuation induced non-exponential kinetics

We show that in experimental atomic force microscopy studies of the lifetime distribution of mechanically stressed folded proteins the effects of externally applied fluctuations can not be distinguished from those of internally present fluctuations. In certain circumstances this leads to artificially non-exponential lifetime distributions which can be misinterpreted as a signature of protein complexity. This work highlights the importance of fully characterizing and controlling external sources of fluctuation in mechanical studies of proteins before drawing conclusions on the physics at play on the molecular level.

I. INTRODUCTION

Atomic force microscopy (AFM) has emerged as a powerful tool with which to study biologically relevant systems [1,2]. Probing the response of a protein to an applied mechanical force allows for the direct investigation of the physical properties of the protein [3,4]. Recent experiments have used lifetime distributions of a single folded protein under constant applied mechanical force to probe physics at the molecular scale [5,6]. To do so, a folded protein is put under tension and the time to unfolding is measured. Many such individual experiments are collected (often over the course of days or even weeks) and their lifetimes combined to estimate the lifetime distribution. Such lifetime distributions have been studied experimentally for both Ubiquitin and the 27th domain of immunoglobulin (I27) [7,8]. In some instances non-exponential behavior has been reported in measurements of Ubiquitin [9,10], and associated with glassy dynamics [9]. Static configurational disorder of the folded protein structure combined with the Bell model [11] has recently been proposed as a plausible explanation for the observed stretched-exponential unfolding time distributions [12,13].

Non-exponential lifetime distributions in physical systems are often a signature of underlying complex processes with multiple timescales operating at the microscopic level [15][16][17]. Characterizing the range of rates present can provide insights on the fundamental physical mechanisms driving the decay process. However, nonexponential distributions can also arise in systems that are characterized by a single timescale, if external fluctuations are able to propagate through non-linearities of the system. Thus, if non-exponential behavior is to be Voltmeter Photodetector Piezo Positioner Cantilever Laser Protein FIG. 1: (Color online) Schematic representation of a typical AFM setup for protein measurements. The deflection of the cantilever is measured as a differential voltage on the quadrant photodetector. In order to achieve a force measurement one must convert the voltage measured on the photodector into a displacement of the tip and then measure the spring constant of the cantilever (typically done using the thermal tune method [14]).

taken as proof of complex internal workings the effects of external fluctuations must be understood.

II. RANDOM BELL MODEL

In his model for cellular adhesion [11], G.I. Bell needed a relationship between rate and mechanical force. He used a phenomenological relation that was found by Zhurkov and Kuksenko in a study of crack propagation in stressed bulk polymers [18]. In analogy with the Arrhenius relation at a given temperature T , this relation postulates an exponential relationship between the un-folding rate α and the applied external force F given by

α = α 0 e βF x ‡ , (1)
where α 0 is the unfolding rate at zero applied force, β = 1/k B T and x ‡ is a model parameter with units of length, usually interpreted as the distance to the transition state [19][20][21]. This relationship was subsequently referred to as the "Bell model" and applied to the dynamic strength of molecular bonds [19] and molecular measurements of protein unfolding by AFMs [22]. This emperical relation is an approximation of the mean escape time from a well (Kramers' problem). The general solution to such a problem depends on the details of the energy landscape and need not take a simple exponential form. Deviations from this approximation will manifest as a force dependent prefactor α 0 (f ) and a force dependent x ‡ (f ) even in the absence of a complex or rough energy landscape [23].

In the application of this model to protein measurements configurational disorder enters as a source of fluctuation for x ‡ and is sufficient to explain experimental results [12,13]. In principle, however, one could consider the effects on the lifetime distribution of fluctuations in all of the parameters, α 0 , β, F and x ‡ .

Fluctuations in α 0 or x ‡ would provide a window into the physics on the microscopic scale [12,13], whereas fluctuations in β and F simply arise from the experimental conditions and provide no insight into protein physics.

We introduce the terminology Random Bell Model to indicate a model similar to the Bell Model (Eq.1) in which each parameter is assumed to be randomly distributed. We will probe this model to determine the impact of various sources of fluctuations on the resulting distributions of rates and lifetimes.

III. SOURCES OF FLUCTUATIONS

Most experiments are done at room temperature, without strict thermal regulation, leading to fluctuations in temperature. However, even in a poorly controlled environment, the upper bound for thermal fluctuations will be on the order of 5°, leading to fluctuations in β of less than 2%. For the sake of brevity we will not consider these fluctuations as being significant. By definition, α 0 is the rate at temperature T and zero applied force, and should be inherent to the folded protein structure. We will therefore assume that α 0 does not fluctuate within a given protein, nor from one molecule to the next. Note that if it proved relevant, it would be straightforward to extend the following results to include fluctuations in both β and α 0 . This leaves us with sources of fluctuations in F and x ‡ to consider. As previously mentioned, fluctuations in x ‡ are the physical quantity of interest which may be masked by fluctations in F . In current experiments many sources FIG. 2: (Color online) Probability density of rate P (α), given by Eq. 11, presented as a Log-Log plot. The curves correspond to values of β F x ‡ = 6.11 and α0 = 10 -3 s -1 , as given in reference [13], for a fixed value of σω = 0. of fluctuations in force are present. Readily visible are the fluctuations in the force value observed over a single experimental trace. These fluctuations include 1. Thermally induced fluctuations [24] in the position and curvature of the cantilever tip as well as the position of the sample.

2. Fluctuations induced at high frequencies by the force feedback system (setup dependent).

3. Mechanical vibrations transmitted through the AFM to the cantilever tip or the sample (setup dependent) [25].

The combined magnitude of these fluctuations has been estimated to be 10% for the apparatus in [10]. Less apparent, but likely more significant, are the fluctuations from experimental run to run induced by the calibration of the system (see Figure 1). These fluctuations include 1. Errors in the calibration of the spring constant of the cantilever (estimates vary from 10% [26] to 20% [27] for the generally used thermal tune method of calibration).

2. Time dependent drift in the spring constant of the AFM cantilever (on the order of 14% in 60 minutes [27]).

Therefore, a worst case estimate of the overall magnitude of the force fluctations would be between 20% and 25%.

IV. UNFOLDING RATE AND TIME DISTRIBUTIONS

A. Unfolding rate distribution

Let F and x ‡ be random variables with respective means F and x ‡ and standard deviations σ F and σ x ‡ . We introduce dimensionless random variables φ and ξ such that

F = φ F , (2)
x ‡ = ξ x ‡ . (3)
These two random variables are described by the joint probability density J(φ, ξ). By definition the probability density of the product ω = φξ is given by

P ω (ω) = +∞ -∞ dφdξJ(φ, ξ)δ(ω -φξ). (4)
To simplify the previous relation, let's perform the following change of variables,

u 1 = φ, (5)
u 2 = φξ, (6)
whose Jacobian is

J = 1 0 ξ φ = |φ| = |u 1 |. (7)
Then Eq.4 becomes

P ω (ω) = +∞ -∞ du 1 du 2 1 |u 1 | J u 1 , u 2 u 1 δ(ω -u 2 ). ( 8
)
The distribution of their product ω = φξ is then given by the Rohatgi formula [START_REF] Rohatgi | An Introduction to Probability and Theory and Mathematical Statistics[END_REF] P

ω (ω) = +∞ -∞ dξ |ξ| J ξ, ω ξ . ( 9
)
This expression of P ω does not require φ and ξ to be independent, nor of any particular distribution. To proceed further, however, we will make the experimentally reasonable assumption that the variables ξ and φ are two independent Gaussian variables, with mean 1 by construction and standard deviations σ ξ = σ x ‡ /x ‡ and σ φ = σ F / F . Note that we do not make any assumption on the relative importance of these two kinds of fluctuations. The distribution of ω is obtained by numerical integration of Eq.9. The distribution of rates α is obtained through the change of variables

α = α 0 exp β F x ‡ ω , (10)
leading to the following probability density of unfolding rates : An example of such a rate distribution is presented in figure 2. We observe a systematic change in the tails of P (α) as the relative weight of σ φ and σ ξ is varied. As these changes are in the tails they may prove difficult to access experimentally.

P α (α) = 1 β F x ‡ α P ω 1 β F x ‡ ln α α 0 . (11)

B. Unfolding time probability distributions

The directly measurable cumulative unfolding time distribution is given by a Laplace transform of the rate distribution as

C(t) = 1 - ∞ 0 dαP α (α)e -αt . ( 12
)
The corresponding probability density is then given by P (t) = dC dt (t). Figures 3a and3c show plots of the cumulative probability distributions and the probability density functions for a fixed value of σ ω , the standard deviation of the product variable ω, and different values of σ φ and σ ξ . The overall shape of the distribution appears to depend only on σ ω : The resulting distributions differ by less than 0.1% and thus are indistinguishable for all values of σ φ and σ ξ for a fixed value of σ ω .

If σ φ very nearly equals zero (i.e. no fluctuations in the externally applied force) no confusion is possible: any deviation from a strictly single exponential lifetime distribution must be attributed to internal fluctuations as described by σ ξ [13]. However this cannot be the case experimentally. Given these sources of fluctuation it is unlikely that σ φ is negligible. Consequently, a good estimate of σ φ must be obtained before one can hope to learn anything about σ ξ or its origins.

C. Skewness of the rate distribution

We have shown that unfolding time statistics are largely insensitive to the origin of the fluctuations. However, we note that the underlying rate distribution becomes asymmetric as the ratio between σ φ and σ ξ grows, as shown in Figure 2. One way to separate the sources of disorder would be experimentally measure the assymetry of the rate distribution as characterized by the skewness. At present such a measurement is experimentally inaccessible because unfolding rates are not experimentally measurable quantities. Instead, they must be estimated from the observed unfolding times using statistical techniques with all of their incumbent limitations, drawbacks, and errors [9,10]. We include the following study of the skewness for two reasons. First, it is the dominant evidence of the separability of internal and external fluctuations. Second, we do so as a way to motivate the development of experimental tools to allow such a measurement to be effected. The skewness γ of the distribution P (α) is defined as

γ = (α -ᾱ) 3 (α -ᾱ) 2 3/2 , ( 13
)
where ᾱ is the mean unfolding rate and • • • stands for the average over P α . Figure 4 presents the evolution of the skewness γ by varying the ratio of σ φ and σ ξ for several fixed values of σ ω , the standard deviation of the product ω = φξ. We observe that the overall scale of the skewness grows with increasing σ ω while the maximum, for fixed σ ω , always occurs when σ φ = σ ξ . Note that skewness is of necessity symmetric under interchange of σ φ and σ ξ because of the symmetry of the original problem. Therefore, γ must have a turning point when σ φ = σ ξ .

Ideally, one should first accurately estimate σ φ , then a measurement of γ will allow for σ ξ to be read off the graph presented in Figure 5. Perhaps more realistically, one can start with a fairly poor estimate of σ φ and then measure γ in a series of experiments in which additional, controlled, amounts of force fluctuation are injected into the experiment to allow for a differential measurement of σ ξ .

V. CONCLUSIONS

In this article we analyzed the effects of both internal and external fluctuations on unfolding time statistics in the context of the random Bell model. We highlighted the extremely weak effect these different origins have on the measurable unfolding time distributions, as illustrated in Fig. 3. Thus it is difficult to attribute non-exponential kinetics solely to the physics of a protein. This observation ultimately comes from the fact that lifetime distribution measurements rely on the assumption that the unfolding times are independent and identically distributed (IID). However, in the presence of external fluctuations this assumption may no longer be justified, particularly if data is gathered over the course of multiple days or weeks using multiple AFM cantilevers with multiple calibrations. Even if the underlying behavior of a protein were perfectly exponential such a lumping together of the data will yield non-exponential lifetime distributions. Statistical analyses, such as the maximum likelihood estimator [START_REF] Aioanei | [END_REF], might provide a way to relax the assumption of IID, at the cost of some arbitrariness in the choice of the likelihood function [START_REF] Brujić | APS Meeting Abstracts[END_REF].

In this article we have focused on the application to constant force protein measurements, however, these arguments can be equally well applied to other afm experiments. As an example, fluctuations in the calibration of experimental setups can have significant consequences on landscape reconstructions using force extension measurements [START_REF] Imparato | [END_REF].

Taken as a whole, these results indicate that it is extremely difficult to experimentally distinguish between internal and external sources of fluctuations by measuring lifetime distributions. External fluctuations can mask the more interesting internal fluctuations. As a necessary but not sufficient prerequisite to measuring the magnitude of internal fluctuations one must first minimize and quantify the level of external fluctuations. Therefore, the internal physics will remain obscured unless AFM force experiments are accompanied by an estimation of the types and magnitudes of external fluctuations and the way that such fluctuations propogate through the measured quantities.

In light of these findings there is a need for new experimental methods in order to gain insight into the physics at play on the molecular scale without relying on a statistical analysis. One promising route is to avoid statistical analysis of collected unfolding time by focusing on protein dynamics. In the spirit of nano-rheology experiments [32,33], direct measurements of the dynamical global response of individual molecules appears as a promising route to understand how interatomic interactions contribute to the mechanical properties of a protein.

FIG. 3 :

 3 FIG. 3: (Color online) Dependence of unfolding time statistics on the relative weight of external fluctuations (σ φ ) and internal fluctuations (σ ξ ) for a fixed value of total fluctuations (σω = 0.20), and for β F x ‡ = 6.11, α0 = 10 -3 s -1 . All four plots are for varying values of (σ φ , σ ξ ) = (0, 0.2) in black solid line, (0.05, 0.19) in orange dotted line, (0.1, 0.17) in red dashed line, (0.15, 0.13) in purple dash dot line. a) Cumulative probability function of unfolding times. The grey line is the best exponential fit. b) Difference between each curve in (a) and the curve for (σ φ , σ ξ ) = (0, σω). c) Probability density function of unfolding times. d) Difference between each curve in (c) and the curve for (σ φ , σ ξ ) = (0, σω). The distributions shown in (a) and (c) are experimentally indistinguishable.

FIG. 4 :FIG. 5 :

 45 FIG.4: (Color online) Evolution of the rate distribution skewness γ as a function of σ φ /σ ξ , for 6 different values of σω, colored from top to bottom as σω = 0.20 (black), 0.19 (purple), 0.18 (brown) 0.17 (red), 0.15 (orange), and 0.10 (yellow) for β F x ‡ = 6.11, α0 = 10 -3 s -1 .

Acknowledgments

We would like to thank Dominique Bicout, Jasna Brujić, Jean-Yves Fortin, Efim Kats, Herbert Lannon, John Royer and Timothy Ziman for discussions and comments.