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On $n$ $n$ -absorbing ideals and $(m, n)$ $(m, n)$ -closed ideals in trivial ring extensions of commutative rings

Department of Mathematics and Statistics, The American University of Sharjah, P. O. Box 26666, Sharjah, United Arab Emirates

E-mail Address: abadawi@aus.edu

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Mohammed Issoual

Department of Mathematics, Faculty of Science and Technology, University S. M. Ben Abdellah, Fez 30000, Morocco

E-mail Address: issoual2@yahoo.fr

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Najib Mahdou

Department of Mathematics, Faculty of Science and Technology, University S. M. Ben Abdellah, Fez 30000, Morocco

E-mail Address: mahdou@hotmail.com

Corresponding author.

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https://doi.org/10.1142/S0219498819501238Cited by:9 (Source: Crossref)

Abstract

Let $R$ $R$ be a commutative ring with $1 \neq 0$ $1 \neq 0$ . Recall that a proper ideal $I$ $I$ of $R$ $R$ is called a 2-absorbing ideal of $R$ $R$ if $a, b, c \in R$ $a, b, c \in R$ and $a b c \in I$ $a b c \in I$ , then $a b \in I$ $a b \in I$ or $a c \in I$ $a c \in I$ or $b c \in I$ $b c \in I$ . A more general concept than 2-absorbing ideals is the concept of $n$ $n$ -absorbing ideals. Let $n \geq 1$ $n \geq 1$ be a positive integer. A proper ideal $I$ $I$ of $R$ $R$ is called an n-absorbing ideal of $R$ $R$ if $a_{1}, a_{2}, \dots, a_{n + 1} \in R$ $a_{1}, a_{2}, \dots, a_{n + 1} \in R$ and $a_{1}, a_{2} \dots a_{n + 1} \in I$ $a_{1}, a_{2} \dots a_{n + 1} \in I$ , then there are $n$ $n$ of the $a_{i}$ $a_{i}$ ’s whose product is in $I$ $I$ . The concept of $n$ $n$ -absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of $R$ $R$ is a 1-absorbing ideal of $R$ $R$ ). Let $m$ $m$ and $n$ $n$ be integers with $1 \leq n < m$ $1 \leq n < m$ . A proper ideal $I$ $I$ of $R$ $R$ is called an $(m, n)$ $(m, n)$ -closed ideal of $R$ $R$ if whenever $a^{m} \in I$ $a^{m} \in I$ for some $a \in R$ $a \in R$ implies $a^{n} \in I$ $a^{n} \in I$ . Let $A$ $A$ be a commutative ring with $1 \neq 0$ $1 \neq 0$ and $M$ $M$ be an $A$ $A$ -module. In this paper, we study $n$ $n$ -absorbing ideals and $(m, n)$ $(m, n)$ -closed ideals in the trivial ring extension of $A$ $A$ by $M$ $M$ (or idealization of $M$ $M$ over $A$ $A$ ) that is denoted by $A (+) M$ $A (+) M$ .

Communicated by E. Gorla

Keywords:

AMSC: 13A15, 13F05, 13G05

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