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arXiv · 2603.18890

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

D. Bennis·A. Bouziri·S. D. Kumar·T. Singh
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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 $S$-$\pi$-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 $\pi$-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.

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D. Bennis, A. Bouziri, S. D. Kumar, T. Singh. 2026-03-19. On the $S$-version of some special elements in commutative rings. https://arxiv.org/abs/2603.18890

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