Finitistic Weak Dimension of Commutative Arithmetical Rings

It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $\leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.

All rings in this paper are unitary and commutative. A ring R is said to be a chain ring if its lattice of ideals is totally ordered by inclusion, and R is called arithmetical if R P is a chain ring for each maximal ideal P . If M is an R-module, we denote by w. When R is an arithmetical ring, by [9], a paper by B. Osofsky, we know that w.gl.d(R) ≤ 1 if R is reduced, and w.gl.d(R) = ∞ otherwise. By using the small finitistic dimension, similar results are proved by S. Glaz and S. Bazzoni when R is a Gaussian ring satisfying one of the following two conditions: • R is locally coherent ([7, Theorem 3.3]); • R contains a prime ideal L such that LR L is nonzero and T-nilpotent (a slight generalization of [2, Theorem 6.4] using [1, Theorems P and 6.3]). They conjectured that these conditions can be removed. Recall that each arithmetical ring is Gaussian.
In this paper, we only investigate the finitistic weak dimension of arithmetical rings (it seems that it is more difficult for Gaussian rings). Main results are summarized in the following theorem: Theorem 1. Let R be an arithmetical ring. Then: (1) f.w.d.(R) = 0 if R is locally IF; (2) f.w.d.(R) = 1 if R is locally semicoherent and not locally IF; ( Let P be a ring property. We say that a ring R is locally P if R P satisfies P for each maximal ideal P . As in [8], a ring R is said to be semicoherent if Hom R (E, F ) is a submodule of a flat R-module for any pair of injective R-modules E, F . A ring R is said to be IF (semi-regular in [8]) if each injective R-module is flat. If R is a chain ring, we denote by P its maximal ideal, Z its subset of zerodivisors which is a prime ideal and Q(= R Z ) its fraction ring. If x is an element of a module M over a ring R, we denote by (0 : x) the annihilator ideal of x and by E(M ) the injective hull of M .
Since flatness is a local module property, Theorem 1 is an immediate consequence of the following theorem that we will prove in the sequel.
Theorem 2. Let R be a chain ring. Then: ( An exact sequence of R-modules 0 → F → E → G → 0 is pure if it remains exact when tensoring it with any R-module. Then, we say that F is a pure submodule of E. When E is flat, it is well known that for any finitely presented R-module F . We recall that a module E is FP-injective if and only if it is a pure submodule of every overmodule.

Proposition 3. A ring R is IF if and only if it is coherent and self FP-injective.
The following proposition will be used frequently in the sequel.
Proposition 4. Let R be a chain ring. The following conditions are equivalent: (1) R is semicoherent; The first assertion of Theorem 2 is an immediate consequence of the following proposition: Let R be a chain ring. Then, for each nonzero element a of P , (0 : a) is a module over Q and it is a flat R/(a)-module.
Proof. Let x ∈ (0 : a) and s a regular element of R. Since R is a chain ring x = sy for some y ∈ R, and 0 = ax = say. It follows that ay = 0. Hence the multiplication by s in (0 : a) is bijective.
If c ∈ R we denote byc the coset c + (a). Any element of (c) ⊗ R/(a) (0 : a) is of the formc ⊗ x where x ∈ (0 : a). Suppose that c / ∈ (a) and cx = 0. There exists t ∈ R such that a = ct = 0. Since R is a chain ring, from cx = 0 and ct = 0 we deduce that x = ty for some y ∈ R. We have ay = cty = cx = 0. So, y ∈ (0 : a) andc ⊗ x =ā ⊗ y = 0. Hence we deduce for each integer q ≥ 1 the following isomorphisms: Proof. Consider the following flat resolution of M : If a ∈ Z we have Tor R p ((0 : a), M ) ∼ = Tor R p+2 (R/(a), M ) = 0 for each integer p ≥ 1. So, the following sequence is exact: is pure-exact over R/(a), and over R too. If 0 = r ∈ Z, there exists 0 = a ∈ Z, such that ra = 0. Hence From these two exact sequences we deduce the following one: It is well known that Q is flat. By We may replace M with a syzygy module of a flat resolution and assume that w.d.(M ) = 3. We consider the following exact sequence where F is flat: By Lemma 8 it remains to prove that w.d.(M ) ≤ 2 if Z ⊗ R M is flat. As above we may assume that R = Q, and by using again the exact sequences (2) and (3)  (1) R/A is IF; (2) either A = P or A is finitely generated; (3) f.w.d.(R/A) = 0.
From these corollaries we deduce the following examples: Example 11. Let R be a valuation domain which is not a field. Let a be a non-zero element of its maximal ideal P . Then R/aR is IF, so f.w.d.(R/aR) = 0. Assume that R contains a non-idempotent prime ideal L. If a is a non-zero element of L, then aL is a non-finitely generated ideal of R L . So, R/aL is not semicoherent and f.w.d.(R/aL) = 2. It is well known that f.w.d.(R) = w.gl.d(R) = 1. Moreover, assume that R contains a non-zero prime ideal L = P . Then, if 0 = a ∈ L then aR L is an ideal of R and R/aR L is semicoherent and not IF. So, f.w.d.(R/aR L ) = 1.
Recall that a chain ring R is strongly discrete if there is no non-zero idempotent prime ideal. From Theorem 1 and [4, Corollary 5.4] we deduce the following: Corollary 12. Let R be an arithmetical ring. The following conditions are equivalent: (1) R is locally strongly discrete; (2) for each proper ideal A, R/A is locally semicoherent; (3) for each proper ideal A, f.w.d.(R/A) ≤ 1.