Local rings of bounded module type are almost maximal valuation rings

It is shown that every commutative local ring of bounded module type is an almost maximal valuation ring

Indecomposable injective modules of finite Malcev rank over local commutative rings

It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $\widehat{R}$ of $R$ is of finite rank over the completion $\widetilde{R}$ of $R$ in the $R$-topology. In this case, for each indecomposable injective module $E$, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar results are obtained for chain rings satisfying some additional properties. It is also shown that each indecomposable injective module over one Krull-dimensional local Noetherian rings has finite Malcev rank. The preservation of Goldie dimension finiteness by localization is investigated too

Finitely presented modules over semihereditary rings

One proves that each almost local-global semihereditary ring has the stacked basis property and is almost Bezout. If M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annhilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a direct sum of 2-generated ideals whose product is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too

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

Valuation domains whose products of free modules are separable

It is proved that if $R$ is a valuation domain with maximal ideal $P$ and if $R_L$ is countably generated for each prime ideal $L$, then $R^R$ is separable if and only $R_J$ is maximal, where $J=\cap_{n\in\mathbb{N}}P^n$

Modules with RD-composition series over a commutative ring

If R is a commutative ring, we prove that every finitely generated module has a pure-composition series with indecomposable factors and any two such series are isomorphic if and only if R is a Bezout ring and a CF-ring

Commutative rings whose cotorsion modules are pure-injective

Let R be a ring (not necessarily commutative). A left R-module is said to be cotorsion if Ext 1 R (G, M) = 0 for any flat R-module G. It is well known that each pure-injective left R-module is cotorsion, but the converse does not hold: for instance, if R is left perfect but not left pure-semisimple then each left R-module is cotorsion but there exist non-pure-injective left modules. The aim of this paper is to describe the class C of commutative rings R for which each cotorsion R-module is pure-injective. It is easy to see that C contains the class of von Neumann regular rings and the one of pure-semisimple rings. We prove that C is strictly contained in the class of locally pure-semisimple rings. We state that a commutative ring R belongs to C if and only if R verifies one of the following conditions: (1) R is coherent and each pure-essential extension of R-modules is essential; (2) R is coherent and each RD-essential extension of R-modules is essential; (3) any R-module M is pure-injective if and only if Ext 1 R (R/A, M) = 0 for each pure ideal A of R (Baer's criterion)

Localization of injective modules over valuation rings

It is proved that the localization of an injective module E, over a valuation ring R, at a prime ideal J, is injective if J is not the subset of zero-divisors of R or if J or E is flat. It follows that localizations of injective modules over h-local Pr\"{u}fer domains are injective too