Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative ring with $ 1 \neq 0$, $I$ a proper ideal of $R$, and $m$ and $n$ positive integers. In this paper, we define $I$ to be a weakly $(m,n)$-closed ideal if $ 0\neq x^{m}\in I$ for $x \in R$ implies $x^{n} \in I$, and $R$ to be an $(m,n)$-von Neumann regular ring if for every $x \in R$, there is an $r \in R$ such that $x^mr = x^n$. A number of results concerning weakly $(m, n)$-closed ideals and $(m,n)$-von Neumann regular rings are given.

Keywords: prime ideal, radical ideal, $2$-absorbing ideal, $n$-absorbing ideal, $(m,n)$-closed ideal, weakly $(m,n)$-closed ideal, $(m,n)$-von Neumann regular

MSC numbers: Primary 13A15; Secondary 13F05, 13G05