Graded generalized 2-absorbing submodules

Let G be a group, R a G-graded commutative ring with nonzero unity, and M a G-graded R-module. In this article, we introduce the concept of graded generalized 2-absorbing submodules as a generalization of graded 2-absorbing submodules, and investigate some properties of this new class of graded submodules. A proper graded R-submodule N of M is said to be a graded generalized 2-absorbing R-submodule of M if whenever x and y are homogeneous elements of R and m is a homogeneous element of M such that xym∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$xym\in N$$\end{document}, then either it is the case that x or y is in the graded radical of (N:Rm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N:_{R} m)$$\end{document}, or xy∈(N:RM)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$xy\in (N:_{R} M)$$\end{document}.