Research ArticleNo Access

On $1$ -absorbing prime ideals of commutative rings

A. Yassine

Faculty of Mathematics, K. N. Toosi University of Technology, P. O. BOX 16315-1618, Tehran, Iran

E-mail Address: yassine_ali@email.kntu.ac.ir

Corresponding author.

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M. J. Nikmehr

Faculty of Mathematics, K. N. Toosi University of Technology, P. O. BOX 16315-1618, Tehran, Iran

E-mail Address: nikmehr@kntu.ac.ir

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, and

R. Nikandish

Department of Mathematics, Jundi-Shapur University of Technology, P. O. BOX 64615-334, Dezful, Iran

E-mail Address: r.nikandish@ipm.ir

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https://doi.org/10.1142/S0219498821501759Cited by:25 (Source: Crossref)

Abstract

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of $1$ -absorbing prime ideals which is a generalization of prime ideals. A proper ideal $I$ of $R$ is called $1$ -absorbing prime if for all nonunit elements $a, b, c \in R$ such that $a b c \in I$ , then either $a b \in I$ or $c \in I$ . Some properties of $1$ -absorbing prime are studied. For instance, it is shown that if $R$ admits a $1$ -absorbing prime ideal that is not a prime ideal, then $R$ is a quasi–local ring. Among other things, it is proved that a proper ideal $I$ of $R$ is $1$ -absorbing prime if and only if the inclusion $I_{1} I_{2} I_{3} \subseteq I$ for some proper ideals $I_{1}, I_{2}, I_{3}$ of $R$ implies that $I_{1} I_{2} \subseteq I$ or $I_{3} \subseteq I$ . Also, $1$ -absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

Communicated by T. H. Ha

Keywords:

AMSC: 13A15, 13C05

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