On the Graded 2-Absorbing Primary Submodules of Graded Multiplication Modules

Fatemeh Soheilnia, Shiroyeh Payrovi

Abstract

Let $G$ be a multiplicative group‎, ‎$R$ be a $G$-graded commutative ring and $M$ be a‎

‎graded $R$-module‎. ‎A proper graded submodule $N$ of $M$ is called graded $2$-absorbing primary‎,

‎if whenever $a,b\in h(R)$ and $m\in h(M)$ with $abm\in N$‎, ‎then $ab\in (N:_R M)$ or‎

‎$am\in Gr_M(N)$ or $bm\in Gr_M(N)$‎. ‎Let $M$ be a graded finitely generated multiplication $R$-module‎.

‎It is shown that $Gr(N‎ :‎_R M) = \big(Gr_M(N)‎ :‎_R M \big)$‎. ‎Furthermore‎, ‎it is proved that $(N‎ :‎_R M)$ is a graded $2$-absorbing primary ideal of $R$‎, ‎if $N$ is a graded $2$-absorbing primary submdoule of $M$‎. ‎Moreover‎, ‎it is generalized some results of graded $2$-absorbing ideals over trivial extension of a ring‎.