The main goal of this article is to explore the concepts of graded $\varphi$-2-absorbing and graded $\varphi$-2-absorbing primary submodules as a new generalization of the concepts of graded 2-absorbing and graded 2-absorbing primary submodules. Let $\varphi :GS\left(M\right)\to GS\left(M\right)\bigcup \left\{\varnothing \right\}$ be a function, where $GS\left(M\right)$ denotes the collection of graded R-submodules of M. A proper $K\in GS\left(M\right)$ is said to be a graded $\varphi$-2-absorbing R-submodule of M if whenever $x,y$ are homogeneous elements of R and s is a homogeneous element of M with $xys\in K-\varphi \left(K\right)$, then $xs\in K$ or $ys\in K$ or $xy\in \left(K{:}_{R}M\right)$, and we call K a graded $\varphi$-2-absorbing primary R-submodule of M if whenever $x,y$ are homogeneous elements of R and s is a homogeneous element of M with $xys\in K-\varphi \left(K\right)$, then $xs$ or $ys$ is in the graded radical of K or $xy\in \left(K{:}_{R}M\right)$. Several properties of these new forms of graded submodules are investigated.

中文翻译：

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分级吸收ϕ-2-和分级吸收-2-初级子模块

本文的主要目标是探索分级的概念 $\varphi$-2-吸收和分级 $\varphi$-2-吸收主子模块是对分2吸收和分2吸收主子模块概念的新概括。让$\varphi ：G\mathrm{小号}（\mathrm{中号}）\to G\mathrm{小号}（\mathrm{中号}）\bigcup \left\{\varnothing \right\}$ 成为一个函数 $G\mathrm{小号}（\mathrm{中号}）$表示M的渐变R-子模块的集合。正确的$ķ\in G\mathrm{小号}（\mathrm{中号}）$ 据说是分级的 $\varphi$-2-吸收ř的-submodule中号如果每当$X，ÿ$是均匀的元素- [R和小号是均匀的元件中号与$Xÿs\in ķ-\varphi （ķ）$， 然后 $Xs\in ķ$ 或者 $ÿs\in ķ$ 或者 $Xÿ\in （ķ{：}_{\mathrm{[R}}\mathrm{中号}）$，我们称K为等级$\varphi$-2-吸收初级ř的-submodule中号如果每当$X，ÿ$是均匀的元素- [R和小号是均匀的元件中号与$Xÿs\in ķ-\varphi （ķ）$， 然后 $Xs$ 或者 $ÿs$在K的渐变根中或$Xÿ\in （ķ{：}_{\mathrm{[R}}\mathrm{中号}）$。研究了这些新形式的渐变子模块的几个属性。