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

Authors : Mohammed ISSOUAL, Najib MAHDOU, Moutu Abdou Salam MOUTUI

Pages : 134-147

Doi:10.24330/ieja.852120

View : 12 | Download : 10

Publication Date : 2021-01-05

Article Type : Research Paper

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 $insert ignore into journalissuearticles values(m, n);$-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 $insert ignore into journalissuearticles values(m,n);$-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 $insert ignore into journalissuearticles values(m,n);$-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