J-IDEALS OF COMMUTATIVE RINGS

Authors : Hani A KHASHAN, Amal B BANIATA

Pages : 148-164

Doi:10.24330/ieja.852139

View : 10 | Download : 7

Publication Date : 2021-01-05

Article Type : Research Paper

Abstract :Let $R$ be a commutative ring with identity and $Ninsert ignore into journalissuearticles values(R);$ and $J\leftinsert ignore into journalissuearticles values(R\right);$ 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 $a\notin Ninsert ignore into journalissuearticles values(R);$, 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\leftinsert ignore into journalissuearticles values(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