A survey on the uniform $S$-version of rings, modules and their homological theories
Xiaolei Zhang, Wei Qi
TL;DR
This survey develops a comprehensive uniform framework for the $S$-version of classical ring and module theory, anchored by a fixed element $s\in S$ to enforce finiteness uniformly across all objects. It introduces and analyzes $u$-$S$-torsion, $u$-$S$-exact sequences, and the uniform analogues of core module classes (notably $u$-$S$-flat, $u$-$S$-projective, $u$-$S$-injective, and $u$-$S$-absolutely pure), along with their homological dimensions. The work establishes broad parallelisms with classical theory, including Cartan-Eilenberg-Bass-type results, local-global characterizations, and change-of-ring theorems, while also developing uniform versions of structural ring classes such as $u$-$S$-von Neumann regular, $u$-$S$-semisimple, and $u$-$S$-Noetherian spectrum. Overall, the paper provides a cohesive, highly structured extension of finiteness and homological concepts to a uniform, $S$-controlled setting, yielding new local-global insights and a robust toolkit for studying polynomial extensions, localizations, and ring constructions under uniform finiteness constraints.
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.