Modules invariant under automorphisms of their covers and envelopes

TL;DR

This paper develops a general theory of modules invariant under automorphisms of their covers and envelopes, extending existing results with clearer proofs across various module types.

Contribution

It introduces a unified framework for modules invariant under automorphisms of their covers and envelopes, simplifying proofs for multiple specific cases.

Findings

Extended and simplified proofs for invariance properties of injective and projective modules.

Established new connections between module invariance and von Neumann regular rings.

Provided a general theory applicable to various module classes.

Abstract

In this paper we develop a general theory of modules which are invariant under automorphisms of their covers and envelopes. When applied to specific cases like injective envelopes, pure-injective envelopes, cotorsion envelopes, projective covers, or flat covers, these results extend and provide a much more succinct and clear proofs for various results existing in the literature. Our results are based on several key observations on the additive unit structure of von Neumann regular rings.