J-IDEALS OF COMMUTATIVE RINGS

Year 2021, Volume: 29 Issue: 29, 148 - 164, 05.01.2021

Hani A. Khashan Amal B. Banı-ata

https://doi.org/10.24330/ieja.852139

Cited By: 14

Abstract

Let $R$ be a commutative ring with identity and $N(R)$ and
$J\left(R\right)$ denote the nilradical and the Jacobson radical
of $R$, respectively. A proper ideal $I$ of $R$ is called an
n-ideal if for every $a,b\in R$, whenever $ab\in I$\ and $a\notin
N(R)$, then $b\in I$. In this paper, we introduce and study
J-ideals as a new generalization of n-ideals in commutative rings.
A proper ideal $I$\ of $R$\ is called a J-ideal if whenever $ab\in
I$\ with $a\notin J\left(R\right) $, then $b\in I$\ for every
$a,b\in R$. We study many properties and examples of such class of
ideals. Moreover, we investigate its relation with some other
classes of ideals such as r-ideals, prime, primary and maximal
ideals. Finally, we, more generally, define and study J-submodules
of an $R$-modules $M$. We clarify some of their properties
especially in the case of multiplication modules.

Keywords

n-ideal, r-ideal, J-ideal, J-submodule

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