Some remarks on almost projective envelopes

TL;DR

This paper studies rings where every module admits an almost projective envelope, exploring their properties and providing characterizations for rings with modules having such envelopes with specific features.

Contribution

It characterizes rings over which all modules have almost projective envelopes and examines conditions for envelopes with special properties.

Findings

Identifies rings where every module has an almost projective envelope.

Provides characterizations for rings with modules having envelopes with certain properties.

Advances understanding of almost projective modules and their envelopes in ring theory.

Abstract

Let $R$ be a commutative ring. An $R$-module $M$ is said to be almost projective if ${\rm Ext}^1_R(M, N) = 0$ for any $R_{\mathfrak{m}}$-module $N$ and any maximal ideal $\mathfrak{m}$ of $R$. In this paper, we investigate rings $R$ over which every $R$-module has an almost projective envelope. We also characterize rings $R$ over which every $R$-module has an almost projective envelope with some special properties.