Graded (m,n)-closed and graded weakly (m,n)-closed ideals

Abstract

Let $G$ be a group with identity $e$ and $R$ a $G$-graded commutative ring with $1\neq 0$. In this paper we introduce the
concept of the graded $(m,n)$-closed ideals and graded weakly $(m,n)$-closed ideals. A graded proper ideal $I$ of $R$ is called a graded $(m,n)$-closed (resp. graded weakly $(m,n)$-closed) ideal if whenever $a^m \in I$ (resp., $0\neq a^m \in I$) for $a \in h(R)$, then $a^n \in I$. Many results are given, in particular we investigate the graded (weakly) $(m,n)$-closed ideals in the direct product $R_1 \times R_2$ of $G$-graded rings $R_1,\ R_2$ and in the trivial extension $R(+)M$ of a $G$-graded ring $R$ by a graded $R$-module $M$.

Keywords:  $G$-graded rings; Trivial extensions, $(m,n)$-closed ideals; Graded $(m,n)$-closed ideals; weakly $(m,n)$-closed ideals; Graded weakly $(m,n)$-closed ideals.

MSC numbers: Primary 13A02, Secondary 13A15, 47B47.

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