On uniformly $S$-absolutely pure modules

TL;DR

This paper introduces and studies $S$-absolutely pure modules and $S$-pure sequences, extending classical concepts, and characterizes certain rings using these modules.

Contribution

It defines $S$-absolutely pure modules and $S$-pure sequences, and characterizes $S$-von Neumann regular and uniformly $S$-Noetherian rings with these modules.

Findings

Characterization of $S$-von Neumann regular rings.

Characterization of uniformly $S$-Noetherian rings.

Extension of classical pure and absolutely pure notions.

Abstract

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we introduce and study the notions of $S$-pure $S$-exact sequences and $S$-absolutely pure modules which extend the classical notions of pure exact sequences and absolutely pure modules. And then we characterize $S$-von Neumann regular rings and uniformly $S$-Noetherian rings using $S$-absolutely pure modules.