Weak dimension of FP-injective modules over chain rings

It is proven that the weak dimension of each FP-injective module over a chain ring which is either Archimedean or not semicoherent is less or equal to 2. This implies that the projective dimension of any countably generated FP-injective module over an Archimedean chain ring is less or equal to 3. By [7, Theorem 1], for any module G over a commutative arithmetical ring R the weak dimension of G is 0, 1, 2 or ∞. In this paper we consider the weak dimension of almost FP-injective modules over a chain ring. This class of modules contains the one of FP-injective modules and these two classes coincide if and only if the ring is coherent. If G is an almost FP-injective module over a chain ring R then its weak dimension is possibly infinite only if R is semicoherent and not coherent. In the other cases the weak dimension of G is at most 2. Moreover this dimension is not equal to 1 if R is not an integral domain. Theorem 15 summarizes main results of this paper. We complete this short paper by considering almost FP-injective modules over local fqp-rings. This class of rings was introduced in [1] by Abuhlail, Jarrar and Kabbaj. It contains the one of arithmetical rings. It is shown the weak dimension of G is infinite if G is an almost FP-injective module over a local fqp-ring which is not a chain ring (see Theorem 23). 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. Chain rings are also called valuation rings (see [9]). If M is an R-module, we denote by w.d.(M) its weak dimension and p.d.(M) its projective dimension. Recall that w.d.(M) ≤ n if TorRn+1(M,N) = 0 for each R-module N . For any ring R, its global weak dimension w.gl.d(R) is the supremum of w.d.(M) where M ranges over all (finitely presented cyclic) R-modules. Its finitistic weak dimension f.w.d.(R) is the supremum of w.d.(M) where M ranges over all R-modules of finite weak dimension. A ring is called coherent if all its finitely generated ideals are finitely presented. As in [14], a ring R is said to be semicoherent if HomR(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 [14]) 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(= RZ) 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 . 2010 Mathematics Subject Classification. 13F30, 13C11, 13E05.