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arXiv · 2512.15084

Structural Analysis of Commutative S-Reduced Rings

Tushar Singh·Shiv Datt Kumar
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Abstract

Let $R$ be a commutative ring with identity, $S \subseteq R$ be a multiplicative set. In this paper, we establish that the intersection of all $S$-prime ideals in an $S$-reduced ring is $S$-zero. Also, we show that an $S$-Artinian reduced ring is isomorphic to the finite direct product of fields. Furthermore, we provide an example of an $S$-reduced ring which is a uniformly-$S$-Armendariz ring (in short, $u$-$S$-Armendariz$)$ ring. Additionally, we prove that the class of uniformly-$S$-reduced rings (in short, $u$-$S$-reduced rings) belongs to the class of $u$-$S$-Armendariz rings. Among other results, we establish the relationship between $S$-reduced rings and $S$-strongly Hopfian rings. Finally, we prove the structure theorem for $S$-reduced rings.

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Tushar Singh, Shiv Datt Kumar. 2025-12-17. Structural Analysis of Commutative S-Reduced Rings. https://arxiv.org/abs/2512.15084

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