On the $S$-version of some special elements in commutative rings

Abstract

In this paper, we introduce and study the $S$-versions of several fundamental elements in commutative rings. Specifically, for a commutative ring $R$ with identity and a multiplicative subset $S$, we define and investigate the notions of $S$-invertible, $S$-idempotent, $S$-von Neumann regular, and $S$-$\ddot{I}€$-regular elements. We establish their basic properties, interrelations, and structural inclusions, and use them to characterize classes of rings. Special attention is given to the uniform $S$-counterparts of Boolean and $\ddot{I}€$-regular rings, where we provide examples distinguishing these from their classical analogues. Several transfer results under homomorphisms and direct product constructions are established, and connections with existing $S$-counterparts (uniformly $S$-von Neumann regular, uniformly $S$-Artinian, etc.) are highlighted. Throughout the paper, we point out several open problems, offering directions for further research.