On the $S$-version of some special elements in commutative rings
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914409240068096 |
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| author | Bennis, D. Bouziri, A. Kumar, S. D. Singh, T. |
| author_facet | Bennis, D. Bouziri, A. Kumar, S. D. Singh, T. |
| contents | 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 $S$-$π$-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 $π$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18890 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the $S$-version of some special elements in commutative rings Bennis, D. Bouziri, A. Kumar, S. D. Singh, T. Commutative Algebra 13A15, 13A99, 13F99 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 $S$-$π$-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 $π$-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. |
| title | On the $S$-version of some special elements in commutative rings |
| topic | Commutative Algebra 13A15, 13A99, 13F99 |
| url | https://arxiv.org/abs/2603.18890 |