SS-Injective Modules and Rings

TL;DR

This paper introduces ss-injectivity as a broad generalization of existing injectivity concepts, providing new characterizations and extending previous results to a wider class of modules and rings.

Contribution

It defines ss-injectivity and strongly ss-injectivity, explores their properties, and offers new characterizations of various important classes of rings, extending prior work on soc-injectivity.

Findings

Characterizations of ss-injective and strongly ss-injective modules and rings

Extension of results on soc-injectivity to ss-injectivity

New criteria for universally mininjective, quasi-Frobenius, Artinian, and semisimple rings

Abstract

We introduce and investigate ss-injectivity as a generalization of both soc-injectivity and small injectivity. A module M is said to be ss-N-injective (where N is a module) if every R-homomorphism from a semisimple small submodule of N into M extends to N. A module M is said to be ss-injective (resp. strongly ss-injective), if M is ss-R-injective (resp. ss-N-injective for every right R-module N). Some characterizations and properties of (strongly) ss-injective modules and rings are given. Some results of Amin, Yuosif and Zeyada on soc-injectivity are extended to ss-injectivity. Also, we provide some new characterizations of universally mininjective rings, quasi-Frobenius rings, Artinian rings and semisimple rings.