Weakly Sigma-cotorsion rings

TL;DR

This paper introduces and studies weakly Sigma-cotorsion rings, characterizing rings where direct sums of injective modules are cotorsion, and explores related classes with finite cotorsion dimension, extending known results.

Contribution

It defines new classes of rings (weakly Sigma-cotorsion and related) and provides new characterizations of n-perfect rings, extending prior research.

Findings

Characterization of weakly Sigma-cotorsion rings.

Extension of results on n-perfect rings.

New criteria for cotorsion dimensions in rings.

Abstract

We study the class of rings $R$ for which every direct sum of injective $R$-modules is cotorsion. We call them weakly $\Sigma$-cotorsion rings. The defining property might be seen as the dual of Chase's characterization of coherence in terms of the flatness of every direct product of projective $R$-modules. More generally, we study rings over which direct sums of injective modules have finite cotorsion dimension and call them weakly $n$-$\Sigma$-cotorsion rings, as well as rings over which direct sums of cotorsion modules have finite cotorsion dimension (called $n$-$\Sigma$-cotorsion rings). In the process, we obtain new characterizations of $n$-perfect rings and extend previous results by Guil Asensio and Herzog, and by \v{S}aroch and \v{S}\v{t}ov\'i\v{c}ek.