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

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