On Generalizations of Graded $2$-absorbing and Graded $2$-absorbing primary submodules

Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a graded unitary $R$-module. In this article, we introduce the concepts of graded $ϕ$-$2$-absorbing and graded $ϕ$-$2$-absorbing primary submodules as generalizations of the concepts of graded $2$-absorbing and graded $2$-absorbing primary submodules. Let $GS(M)$ be the set of all graded $R$-submodules of $M$ and $ϕ: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded $ϕ$-$2$-absorbing $R$-submodule of $M$ if whenever $x, y$ are homogeneous elements of $R$ and $m$ is a homogeneous element of $M$ with $xym\in K-ϕ(K)$, then $xm\in K$ or $ym\in K$ or $xy\in (K :_{R} M)$, and $K$ is said to be a graded $ϕ$-$2$-absorbing primary $R$-submodule of $M$ if whenever $x, y$ are homogeneous elements of $R$ and $m$ is a homogeneous element of $M$ with $xym\in K-ϕ(K)$, then $xm$ or $ym$ is in the graded radical of $K$ or $xy\in (K:_{R}M)$. We investigate several properties of these new types of graded submodules.

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