AI Summary & Key Takeaways
AI InsightThis 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.
📌 3 Key Takeaways
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.