Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2023; 38(4): 993-999

Online first article October 16, 2023      Printed October 31, 2023

Copyright © The Korean Mathematical Society.

On graded $(m, n)$-closed submodules

Rezvan Varmazyar

Khoy Branch, Islamic Azad University


Let $A$ be a $G$-graded commutative ring with identity and $M$ a graded $A$-module. Let $m, n$ be positive integers with $m>n$. A proper graded submodule $L$ of $M$ is said to be graded $(m, n)$-closed if $a^{m}_g\cdot x_t\in L$ implies that $a^{n}_g\cdot x_t\in L$, where $a_g\in h(A)$ and $x_t\in h(M)$. The aim of this paper is to explore some basic properties of these class of submodules which are a generalization of graded $(m, n)$-closed ideals. Also, we investigate $GC^{m}_n-rad$ property for graded submodules.

Keywords: Graded submodule, graded $(m, n)$-closed, graded radical

MSC numbers: 13A02, 16W50

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