Weakly Sigma-cotorsion rings

TL;DR

The paper investigates rings for which direct sums of injective modules exhibit cotorsion behavior, introducing the notion of weakly $Σ$-cotorsion rings and its $n$-parameterized generalizations. It develops a robust framework of $n$-$Σ$-cotorsion modules and rings, establishing multiple equivalent conditions for left $n$-perfection and proving dual characterizations to Chase’s coherence criteria, including definable closures and cotorsion envelopes. A key theme is the closure properties of $ ext{Cot}_n$ under direct sums and limits, and the corresponding injective-side dualities that relate to FP-injective and pure-injective modules. The paper also investigates triangular matrix rings, providing concrete transfer results: $T=egin{pmatrix} A & 0\ U & B \\ abla abla abla abla\

Abstract

We study the class of rings $R$ for which every direct sum of injective $R$-modules is cotorsion. We call them weakly $Σ$-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 $n$-$Σ$-cotorsion rings, as well as rings over which direct sums of cotorsion modules have finite cotorsion dimension (called $n$-$Σ$-cotorsion rings). In the process, we obtain new characterizations of $n$-perfect rings and extend previous results by Guil Asensio and Herzog, and by Šaroch and Šťovíček.

Theorems & Definitions (29)

• Definition 3.1
• Lemma 3.2
• Definition 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Theorem 3.7
• Definition 4.1
• Theorem 4.2
• Remark 4.3