ON $(m,n)$-CLOSED IDEALS IN AMALGAMATED ALGEBRA

Year 2021, Volume: 29 Issue: 29, 134 - 147, 05.01.2021

Mohammed Issoual Najib Mahdou Moutu Abdou Salam Moutuı

https://doi.org/10.24330/ieja.852120

Cited By: 2

Abstract

Let $R$ be a commutative ring with $1 \ne 0$ and let $m$ and
$n$ be integers with $1\leq n < m$. A proper ideal $I$ of $R$ is
called an $(m, n)$-closed ideal of $R$ if whenever $a^m \in I$ for
some $a\in R$ implies $a^n \in I$. Let $ f:A\rightarrow B$ be a
ring homomorphism and let $J$ be an ideal of $B.$ This paper
investigates the concept of $(m,n)$-closed ideals in the
amalgamation of $A$ with $B$ along $J$ with respect $f$ denoted by
$A\bowtie^{f}J$. Namely, Section 2 investigates this notion to
some extensions of ideals of $A$ to $A\bowtie^fJ$. Section 3
features the main result, which examines when each proper ideal of
$A\bowtie^fJ$ is an $(m,n)$-closed ideal. This allows us to give
necessary and sufficient conditions for the amalgamation to
inherit the radical ideal property with applications on the
transfer of von Neumann regular, $\pi$-regular and semisimple
properties.

Keywords

$(m,n)$-closed ideal, radical ideal, semi-$n$-absorbing ideal, amalgamated algebra, von Neumann regular ring, $\pi$-regular ring, semisimple ring

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