Commutative rings whose cotorsion modules are pure-injective

TL;DR

This paper characterizes commutative rings where every cotorsion module is pure-injective, revealing structural conditions that distinguish such rings from broader classes like locally pure-semisimple rings.

Contribution

It provides a complete description of commutative rings with the property that cotorsion modules are always pure-injective, including necessary and sufficient conditions.

Findings

Class C contains von Neumann regular and pure-semisimple rings.

Class C is strictly contained within locally pure-semisimple rings.

R belongs to C if and only if it satisfies specific coherence and extension conditions.

Abstract

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).