Subinjective and Subprojective Extension-Reflecting Modules

TL;DR

This paper investigates conditions under which classes of modules related to subinjectivity and subprojectivity are closed under extensions, providing new characterizations and examples over various rings, including QF-rings.

Contribution

It establishes criteria ensuring extension-closure of subinjective and subprojective classes, especially when modules have injective hulls that are projective or are homomorphic images of modules with combined properties.

Findings

Extension-closure of rIn(M) when injective hull of M is projective.

Extension-closure of rPr(M) when M is a homomorphic image of a module that is both projective and injective.

Over QF-rings, rIn(M) and rPr(M) are always extension-closed.

Abstract

Given a right R-module M and any short exact sequence of right R-modules \[ 0 \to A \to B \to C \to 0, \] it is well known that if both A and C belong to the subinjectivity domain $\mathfrak{\underline{In}^{-1}}(M)$ (resp., the subprojectivity domain $\mathfrak{\underline{Pr}^{-1}}(M)$) of M, then B also belongs to the corresponding domain. Module classes satisfying this closure property are said to be closed under extensions. Let $ \mathfrak{In}(M) = \{ N \in Mod-R \mid M \in \mathfrak{\underline{In}^{-1}}(N) \}$ and $ \mathfrak{Pr}(M) = \{ N \in Mod-R \mid M \in \mathfrak{\underline{Pr}^{-1}}(N)\}.$ Unlike $\mathfrak{\underline{In}^{-1}}(M)$ and $\mathfrak{\underline{Pr}^{-1}}(M)$, the classes $ \mathfrak{In}(M)$ and $\mathfrak{Pr}(M) $ are not, in general, closed under extensions. In this paper, we investigate certain classes of modules and rings that ensure the extension-closure of $ \mathfrak{In}(M)$ and $\mathfrak{Pr}(M) $. We prove that if the injective hull of $M$ is projective, then $ \mathfrak{In}(M)$ is closed under extension; and if $M$ is a homomorphic image of a module which is both projective and injective, then $\mathfrak{Pr}(M) $. is closed under extensions. As a consequence, over a QF-ring, the classes $ \mathfrak{In}(M)$ and $\mathfrak{Pr}(M) $ are closed under extensions for every module $M$. We further explore several implications of these results and present examples of modules with the above properties over arbitrary rings. Additionally, we introduce the rings whose right modules of finite length are homomorphic images of injective modules. Among other results, we prove that, if $ \mathfrak{In}(R)$ is closed under extensions, then modules of finite length are homomorphic image of injectives if and and only if simple modules are homomorphic image of injectives.