Subinjective and Subprojective Extension-Reflecting Modules

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.