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