A survey on the uniform $S$-version of rings, modules and their homological theories

TL;DR

This survey reviews recent developments in the theory of uniformly $S$-algebraic structures, extending classical ring and module concepts with a focus on uniform $S$-analogues and their homological properties.

Contribution

It systematically introduces and characterizes the notion of uniform $S$-structures, providing a comprehensive framework for their modules, homological dimensions, and structural properties.

Findings

Defined and characterized $u$-$S$-torsion modules and $u$-$S$-exact sequences.

Explored uniform homological dimensions and their relations with polynomial rings and localizations.

Analyzed structural ring classes like $u$-$S$-von Neumann regular and $u$-$S$-Artinian rings.

Abstract

This survey provides a comprehensive overview of the recent advancements in the theory of ``uniformly $S$''-algebraic structures in commutative ring theory. Originating from the classical concepts of Noetherian, coherent, von Neumann regular, and semisimple rings, the introduction of a multiplicative subset $S$ has led to the development of $S$-Noetherian, $S$-coherent, and other $S$-analogues. However, the element $s \in S$ in the original definitions often depends on the ideal or module under consideration. To overcome this limitation and enable deeper module-theoretic characterizations, the notion of "uniformly $S$" (abbreviated as $u$-$S$) was introduced. This survey systematically presents the definitions, characterizations, and properties of $u$-$S$-torsion modules, $u$-$S$-exact sequences, and the subsequent uniform analogues of fundamental module classes: $u$-$S$-finitely presented, $u$-$S$-Noetherian, $u$-$S$-coherent, $u$-$S$-flat, $u$-$S$-projective, $u$-$S$-injective, and $u$-$S$-absolutely pure modules. We then explore the associated uniform homological dimensions, including the $u$-$S$-weak global dimension, the $u$-$S$-global dimension, and their interplay with polynomial rings and localizations. The survey also covers structural ring classes such as $u$-$S$-von Neumann regular, $u$-$S$-semisimple, $u$-$S$-Artinian, $u$-$S$-multiplication rings, and rings with $u$-$S$-Noetherian spectrum.