S-r-ideals in commutative rings
DOI:
https://doi.org/10.5269/bspm.67878Abstract
The rings considered in this article are commutative with identity. This article is motivated by the results proved by Visweswaran (\cite{S-primary}) on $S$-primary ideals. In this paper, we introduce the concept of $S$-$r$-ideal (resp., $S$-$pr$-ideal) of a commutative ring and study its properties. Let $R$ be a commutative ring with $1\neq0$ and $S$ be a multiplicatively closed subset of $R$. Let $I$ be an ideal of $R$ disjoint with $S.$ We say that $I$ is an $S$-{\it r}-ideal (resp., $S$-{\it pr}-ideal) of $R$ if there exists an $s \in S$ such that for all $a, b \in R$ if $ ab \in I$ with $Ann(a) = (0)$ implies that $sb\in I$ (resp., $sb \in \sqrt{I}$). We investigate many properties and characterizations of $S$-$r$-ideals (resp., $S$-$pr$-ideals). We discuss the form of $S$-$r$-ideals (resp., $S$-$pr$-ideals) in a finite direct product of rings. Furthermore, we study $S$-$r$-ideals (resp., $S$-$pr$-ideals) in Nagata's idealization ring. Our results allow us to construct original examples of $S$-$r$-ideals (resp., $S$-$pr$-ideals).
References
[2] Anderson, D.D., Winders, M. , Idealization of a module. J. Commut. Algebra. 1 (1), 3-56,(2009).
[3] Atani, S.E, Farzalipour, F. , On weakly primary ideals. GMJ. 12 (3): 423-429, (2005).
[4] Hamed, A., Malek, A. , S-prime ideals of a commutative ring. Beitr. Algebra Geom. 61 (3), 533-542,(2020).
[5] Gilmer, R. , Multiplicative Ideal Theory. Dekker, New York,(1972).
[6] Kaplansky, I. ,Commutative Rings. rev. ed., Univ. Chicago Press, Chicago,(1974).
[7] Massaoud, E. , S-primary ideals of a commutative ring. Commun. Algebra. 50(3),988-997,(2022).
[8] Mohamadian, R. , r-ideals in commutative rings. Turk. J. Math. 39,733-749,(2015).
[9] Uregen, R. N., On uniformly pr-ideals in commutative rings. Turk. J. Math. 43,1878-1886, (2019).
[10] Visweswaran, S. , Some results on S-primary ideals of a commutative ring. Beitr Algebra Geom. 63(2), 247-266, (2021)
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