Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules

Abstract

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 $\varphi :GS\left(M\right)\to GS\left(M\right)\bigcup \{\varnothing \}$ 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.

1. Introduction

Throughout this article, G will be a group with identity e and R a commutative ring with nonzero unity 1. Then R is said to be G-graded if ${\displaystyle R=\underset{g\in G}{⨁}{R}_g}$ with $R_g{R}_h\subseteq R_{gh}$ for all $g,h\in G$ where $R_g$ is an additive subgroup of R for all $g\in G$. The elements of $R_g$ are called homogeneous of degree g. If $x\in R$, then x can be written uniquely as ${\displaystyle \sum _{g\in G}{x}_g}$, where $x_g$ is the component of x in $R_g$. It is known that $R_e$ is a subring of R and $1\in R_e$. The set of all homogeneous elements of R is ${\displaystyle h\left(R\right)=\bigcup _{g\in G}{R}_g}$. Assume that M is a left unitary R-module. Then M is said to be G-graded if ${\displaystyle M=\underset{g\in G}{⨁}{M}_g}$ with $R_g{M}_h\subseteq M_{gh}$ for all $g,h\in G$ where $M_g$ is an additive subgroup of M for all $g\in G$. The elements of $M_g$ are called homogeneous of degree g. It is clear that $M_g$ is an $R_e$-submodule of M for all $g\in G$. If $x\in M$, then x can be written uniquely as ${\displaystyle \sum _{g\in G}{x}_g}$, where $x_g$ is the component of x in $M_g$. The set of all homogeneous elements of M is ${\displaystyle h\left(M\right)=\bigcup _{g\in G}{M}_g}$. Let K be an R-submodule of a graded R-module M. Then K is said to be graded R-submodule if ${\displaystyle K=\underset{g\in G}{⨁}(K\cap M_g)}$, i.e., for $x\in K$, ${\displaystyle x=\sum _{g\in G}{x}_g}$ where $x_g\in K$ for all $g\in G$. An R-submodule of a graded R-module need not be graded. For more details and terminology, see [1,2].

Specifically, $An{n}_R\left(M\right)=\left(0{:}_{R}M\right)$ turns out to be a graded ideal of R. May I be an ideal of R, rightly graded. Then the graded radical of I will be $Grad\left(I\right)$, and is defined to be the collection of all $r\in R$ so that for every $g\in G$, there exists $n_g\in \mathbb{N}$ for which $r_g^{n_g}\in I$. One can see that if $r\in h\left(R\right)$, then $r\in Grad\left(I\right)$ if and only if $r^n\in I$ for some $n\in \mathbb{N}$. In fact, $Grad\left(I\right)$ is a graded ideal of R, see [3]. Let K be a graded R-submodule of M. The graded radical of K is then denoted by $Gra{d}_M\left(K\right)$ and is specified as the intersection of all graded prime submodules of M enclosing K. If no graded prime submodule contains K exists, we can take $Gra{d}_M\left(K\right)=M$.

A graded prime (resp. graded primary) R-submodule is a graded R-submodule $K\ne M$ with the property that for $a\in h\left(R\right)$ and $s\in h\left(M\right)$ such that $as\in K$ implies that $s\in K$ or $a\in \left(K{:}_{R}M\right)$ (resp. $s\in K$ or $a\in Grad\left(\left(K{:}_{R}M\right)\right))$. As graded prime ideals (submodules) have an important role in graded ring (module) theory, several authors generalized these concepts in different ways, see ([4,5,6,7,8]). Atani in [9] has introduced graded weakly prime submodules. A graded R-submodule $K\ne M$ is called graded weakly prime if whenever $a\in h\left(R\right)$ and $s\in h\left(M\right)$ with $0\ne as\in K$, then $s\in K$ or $a\in \left(K{:}_{R}M\right)$. The concept of graded $\varphi$-prime submodules has been introduced in [10]. Let $\varphi :GS\left(M\right)\to GS\left(M\right)\bigcup \{\varnothing \}$ be a function, where $GS\left(M\right)$ denotes the collection of graded submodules of M. A graded R-submodule $K\ne M$ is called graded $\varphi$-prime if whenever $a\in h\left(R\right)$ and $s\in h\left(M\right)$ such that $as\in K-\varphi \left(K\right)$, then $s\in K$ or $a\in \left(K{:}_{R}M\right)$.

Graded 2-absorbing ideals (resp. graded weakly 2-absorbing ideals) are introduced in [11] as a different generalization of graded prime ideals (resp. graded weakly prime ideals). A proper graded ideal I of R is a graded 2-absorbing ideal (resp. graded weakly 2-absorbing ideal) of R if whenever $x,y,z\in h\left(R\right)$ and $xyz\in I$ (resp. $0\ne xyz\in I$), then $xy\in I$ or $xz\in I$ or $yz\in I$. Then introducing graded 2-absorbing submodules (resp. graded weakly 2-absorbing submodules) in [5] generalized the concept of graded 2-absorbing ideals (resp. graded weakly 2-absorbing ideals) to graded submodules.

Al-Zoubi and Sharafat in [12] proposed the notion of graded 2-absorbing primary ideals, where a proper graded ideal I of R is called graded 2-absorbing primary if whenever $x,y,z\in h\left(R\right)$ with $xyz\in I$, then $xy\in I$ or $xz\in Grad\left(I\right)$ or $yz\in Grad\left(I\right)$. The notion of graded 2-absorbing primary submodules is studied in [7] as a generalization of graded 2-absorbing primary ideals. A proper graded R-submodule K of M is said to be a graded 2-absorbing primary R-submodule (resp. graded weakly 2-absorbing primary R-submodule) of M if whenever $x,y\in h\left(R\right)$ and $m\in h\left(M\right)$ with $xym\in K$ (resp. $0\ne xym\in K$), then $xy\in \left(K{:}_{R}M\right)$ or $xm\in Gra{d}_M\left(K\right)$ or $ym\in Gra{d}_M\left(K\right)$.

A graded R-module M is called a graded multiplication if for any graded R-submodule K, there exists a graded ideal I of R with $K=IM$. In this instance, $I=\left(K{:}_{R}M\right)$. Graded multiplication modules were firstly introduced by Escoriza and Torrecillas in [13], and further results were obtained by several authors, see for example [14]. Let N and K be graded R-submodules of a graded multiplication R-module M with $N=IM$ and $K=JM$ for some graded ideals I and J of R. The product of N and K is denoted by $NK$ is defined by $NK=IJM$. Then the product of N and K is independent of presentations of N and K. In fact, as $IJ$ is a graded ideal of R (see [2]), $NK$ is a graded R-submodule of M and $NK\subseteq N\bigcap K$. Moreover, for $x,y\in h\left(M\right)$, by $xy$, we mean the product of $Rx$ and $Ry$. Also, it can be seen in ([8], Theorem 9) that if N is a proper graded R-submodule of a graded multiplication R-module M, then $Gra{d}_M\left(N\right)=Grad\left(\left(N{:}_{R}M\right)\right)M$.

In this article, our aim is to extend the notion of graded 2-absorbing submodules to graded $\varphi$-2-absorbing submodules using similar techniques to that are used in [10], and also to extend graded 2-absorbing primary submodules to graded $\varphi$-2-absorbing primary submodules. Our study is inspired by [15].

2. Graded $\mathit{\varphi }$-2-Absorbing and Graded $\mathit{\varphi }$-2-Absorbing Primary Submodules

This segment includes proposing and examining the notions of graded $\varphi$-2-absorbing and graded $\varphi$-2-absorbing primary submodules.

Definition 1.

Let M be a G-graded R-module and $\varphi :GS\left(M\right)\to GS\left(M\right)\bigcup \{\varnothing \}$ be a function.

1. 

A graded R-submodule $K\ne M$ is called graded ϕ-primary submodule if whenever $a\in h\left(R\right)$ and $s\in h\left(M\right)$ with $as\in K-\varphi \left(K\right)$, then $s\in K$ or $a\in Grad\left(\left(K{:}_{R}M\right)\right)$.

2. 

A graded R-submodule K with $K_g\ne M_g$ for some $g\in G$ is called g-ϕ-primary submodule if whenever $a\in R_e$ and $s\in M_g$ with $as\in K-\varphi \left(K\right)$, then $s\in K$ or $a\in Grad\left(\left(K{:}_{R}M\right)\right)$.

3. 

A graded R-submodule $K\ne M$ is called graded ϕ-2-absorbing submodule (ϕ-2-abs. submodule) if whenever $x,y\in h\left(R\right)$ and $s\in h\left(M\right)$ with $xys\in K-\varphi \left(K\right)$, then $xs\in K$ or $ys\in K$ or $xy\in \left(K{:}_{R}M\right)$.

4. 

A graded R-submodule K with $K_g\ne M_g$ for some $g\in G$ is called g-ϕ-2-absorbing submodule (g-ϕ-2-abs. submodule) if whenever $x,y\in R_e$ and $s\in M_g$ with $xys\in K-\varphi \left(K\right)$, then $xs\in K$ or $ys\in K$ or $xy\in \left(K{:}_{R}M\right)$.

5. 

A graded R-submodule $K\ne M$ of M is called graded ϕ-2-absorbing primary submodule (ϕ-2-abs. prim. submodule) if whenever $x,y\in h\left(R\right)$ and $s\in h\left(M\right)$ with $xys\in K-\varphi \left(K\right)$, then $xs\in Gra{d}_M\left(K\right)$ or $ys\in Gra{d}_M\left(K\right)$ or $xy\in \left(K{:}_{R}M\right)$.

6. 

A graded R-submodule K with $K_g\ne M_g$ for some $g\in G$ is called g-ϕ-2-absorbing primary submodule (g-ϕ-2-abs. prim. submodule) if whenever $x,y\in R_e$ and $s\in M_g$ with $xys\in K-\varphi \left(K\right)$, then $xs\in Gra{d}_M\left(K\right)$ or $ys\in Gra{d}_M\left(K\right)$ or $xy\in \left(K{:}_{R}M\right)$.

7. 

A proper graded R-submodule K of M is said to be a graded ϕ-almost primary R-submodule of M if whenever $r\in h\left(R\right)$ and $m\in h\left(M\right)$ such that $rm\in K-\varphi \left(K\right)$, then either $m\in K$ or $r\in Gra{d}_M\left(K\right)$.

Remark 1.

1. 

For a graded ϕ-primary R-submodule K of a graded multiplication R-module M, we have the following functions:

${\varphi }_{\varnothing }\left(K\right)=\varnothing$ graded primary submodule,

${\varphi }_0\left(K\right)=\left\{0\right\}$ graded weakly primary submodule,

${\varphi }_2\left(K\right)=K^2$ graded almost primary submodule,

${\varphi }_n\left(K\right)=K^n$ graded n-almost primary submodule, and

${\displaystyle {\varphi }_{\omega }\left(K\right)=\bigcap _{n=1}^{\infty }{K}^n}$ graded ω-primary submodule.

2. 

Let K be a graded ϕ-2-abs. R-submodule (resp. graded ϕ-2-abs. prim. R-submodule) of a graded multiplication R-module M. Then

${\varphi }_{\varnothing }\left(K\right)=\varnothing$ graded 2-abs. submodule (resp. graded 2-abs. prim. submodule),

${\varphi }_0\left(K\right)=\left\{0\right\}$ graded weakly 2-abs. submodule (resp. graded weakly 2-abs. prim. submodule),

${\varphi }_2\left(K\right)=K^2$ graded almost 2-abs. submodule (resp. graded almost 2-abs. prim. submodule),

${\varphi }_n\left(K\right)=K^n$ graded n-almost 2-abs. submodule (resp. graded n-almost 2-abs. prim. submodule), and

${\displaystyle {\varphi }_{\omega }\left(K\right)=\bigcap _{n=1}^{\infty }{K}^n}$ graded ω-2-abs. submodule (resp. graded ω-2-abs. prim. submodule).

3. 

For functions $\varphi ,\psi :GS\left(M\right)\to GS\left(M\right)\bigcup \{\varnothing \}$, we write $\varphi \le \psi$ if $\varphi \left(K\right)\subseteq \psi \left(K\right)$ for all $K\in GS\left(M\right)$. Obviously, therefore, we have the next order:

${\psi }_{\varnothing }\le {\psi }_0\le {\psi }_{\omega }\le ...\le {\psi }_{n+1}\le {\psi }_n\le ...\le {\psi }_2\le {\psi }_1$.

4. 

If $\varphi \le \psi$, then every graded ϕ-2-abs. submodule (resp. graded ϕ-2-abs. prim. submodule) is graded ψ-2-abs. submodule (resp. graded ψ-2-abs. prim.submodule).

Remark 2.

Since $K-\varphi \left(K\right)=K-(K\bigcap \varphi (K\left)\right)$ for any graded R-submodule K, we may assume $\varphi \left(K\right)\subseteq K$ without loss of generality, and we will do so throughout this article.

Theorem 1.

Let M be a graded R-module and K be a proper graded R-submodule of M. Then the following implications hold:

1. 

K is graded ϕ-prime submodule ⇒K is graded ϕ-2-abs. submodule ⇒K is graded ϕ-2-abs. prim. submodule.

2. 

If M is a graded multiplication R-module and K is a graded ϕ-primary submodule, then K is graded ϕ-2-abs. prim. submodule

3. 

For graded multiplication R-module M, K is graded 2-abs. submodule ⇒K is graded weakly 2-abs. submodule ⇒K is graded ω-2-abs. submodule ⇒K is graded $(n+1)$-almost 2-abs. submodule ⇒K is graded n-almost 2-abs. submodule for all $n\ge 2$⇒K is graded almost 2-abs. submodule.

4. 

For graded multiplication R-module M, K is graded 2-abs. prim. submodule ⇒K is graded weakly 2-abs. prim. submodule ⇒K is graded ω-2-abs. prim. submodule ⇒K is graded $(n+1)$-almost 2-abs. prim. submodule ⇒K is graded n-almost 2-abs. prim. submodule for all $n\ge 2$⇒K is graded almost 2-abs. prim. submodule.

5. 

Suppose that $Gra{d}_M\left(K\right)=K$. Then K is graded ϕ-2-abs. prim. submodule if and only if K is graded ϕ-2-abs. submodule.

6. 

If M is a graded multiplication R-module and K is an idempotent graded R-submodule of M, then K is graded ω-2-abs. submodule, and K is graded n-almost 2-abs. submodule for every $n\ge 2$.

7. 

Let M be a graded multiplication R-module. Then K is graded n-almost 2-abs. submodule (resp. graded n-almost 2-abs. prim. submodule) for all $n\ge 2$ if and only if K is graded ω-2-abs. submodule (resp. graded ω-2-abs. prim. submodule).

Proof. 

• This is straightforward.
• Let $x,y\in h\left(R\right)$ and $m\in h\left(M\right)$ with $xym\in K-\varphi \left(K\right)$. suppose that $ym\notin Gra{d}_M\left(K\right)$. Then $ym\notin K$ and then $x\in Grad\left(\left(K{:}_{R}M\right)\right)$ as K is a graded $\varphi$-primary R-submodule. Therefore, $xm\in Grad(\left(K{:}_{R}M\right)M=Gra{d}_M\left(K\right)$. Consequently, K is graded $\varphi$-2-absorbing primary.
• This is clear by Remark 1 (4).
• This is clear by Remark 1 (4).
• The claim is obvious.
• Since K is an idempotent R-submodule, $K^n=K$ for all $n>0$, and then ${\displaystyle {\varphi }_{\omega }\left(K\right)=\bigcap _{n=1}^{\infty }{K}^n=K}$. Thus K is a graded $\omega$-2-abs.. By (3), we conclude that K is a graded n-almost 2-abs. for all $n\ge 2$.
• Suppose that K is a graded n-almost 2-absorbing (resp. graded n-almost 2-absorbing primary) R-submodule of M for all $n\ge 2$. Let $x,y\in h\left(R\right)$ and $m\in h\left(M\right)$ with $xym\in K$ but ${\displaystyle xym\notin \bigcap _{n=1}^{\infty }{K}^n}$. Hence $xym\notin K^n$ for some $n\ge 2$. Since K is graded n-almost 2-abs. (resp. graded n-almost 2-abs. prim.) for all $n\ge 2$, this implies either $xy\in \left(K{:}_{R}M\right)$ or $ym\in K$ or $xm\in K$ (resp. $xy\in \left(K{:}_{R}M\right)$ or $ym\in Gra{d}_M\left(K\right)$ or $xm\in Gra{d}_M\left(K\right)$). This completes the first implication. The converse is clear from (3) (resp. from (4)).

□

Proposition 1.

Let M be a graded R-module and K be a graded ϕ-2-abs. prim. submodule of M. If $\varphi \left(K\right)$ is a graded 2-abs. prim. submodule of M, then K is a graded-2-abs. prim. submodule of M.

Proof. 

Let $x,y\in h\left(R\right)$ and $m\in h\left(M\right)$ such that $xym\in K$. If $xym\in \varphi \left(K\right)$, then we conclude that $xm\in Gra{d}_M\left(\varphi \left(K\right)\right)\subseteq Gra{d}_M\left(K\right)$ or $ym\in Gra{d}_M\left(\varphi \left(K\right)\right)\subseteq Gra{d}_M\left(K\right)$ or $xy\in \left(\varphi \left(K\right){:}_{R}M\right)\subseteq \left(K{:}_{R}M\right)$ since $\varphi \left(K\right)$ is graded 2-abs. prim., and so the result holds. If $xym\notin \varphi \left(K\right)$, then the result holds easily since K is graded $\varphi$-2-abs. prim. submodule. □

Theorem 2.

Let M be a graded R-module and K be a graded R-submodule of M with $\varphi \left(Gra{d}_M\left(K\right)\right)$$\subseteq \varphi \left(K\right)$. If $Gra{d}_M\left(K\right)$ is a graded ϕ-prime submodule of M, then K is a graded ϕ-2-abs. prim. submodule of M.

Proof. 

Let $x,y\in h\left(R\right)$ and $m\in h\left(M\right)$ with $xym\in K-\varphi \left(K\right)$ and $xm\notin Gra{d}_M\left(K\right)$. Because $Gra{d}_M\left(K\right)$ is a graded $\varphi$-prime and $xym\in Gra{d}_M\left(K\right)-\varphi \left(Gra{d}_M\left(K\right)\right)$, $y\in \left(Gra{d}_M\left(K\right){:}_{R}M\right)$. So, $ym\in Gra{d}_M\left(K\right)$. Consequently, K is a graded $\varphi$-2-abs. prim. of M. □

Next we consider the behaviour of graded $\varphi$-2-absorbing (primary) submodules under graded quotient modules. Let M be a G-graded R-module and K be a graded R-submodule of M. Then $M/K$ is G-graded by ${(M/K)}_g=(M_g+K)/K$ for all $g\in G$ ([2]).

Lemma 1

([16], Lemma 3.2). Let M be a graded R-module, K be a graded R-submodule of M, and N be an R-submodule of M such that $K\subseteq N$. Then N is a graded R-submodule of M if and only if $N/K$ is a graded R-submodule of $M/K$.

Theorem 3.

Let M be a graded R-module and K be a proper graded R-submodule of M.

1. 

K is a graded ϕ-2-abs. submodule of M if and only if $K/\varphi \left(K\right)$ is a graded weakly 2-abs. submodule of $M/\varphi \left(K\right)$.

2. 

K is a graded ϕ-2-abs. prim. submodule of M if and only if $K/\varphi \left(K\right)$ is a graded weakly 2-abs. prim. submodule of $M/\varphi \left(K\right)$.

3. 

K is a graded ϕ-prime submodule of M if and only if $K/\varphi \left(K\right)$ is a graded weakly prime submodule of $M/\varphi \left(K\right)$.

4. 

K is a graded ϕ-primary submodule of M if and only if $K/\varphi \left(K\right)$ is a graded weakly primary submodule of $M/\varphi \left(K\right)$.

Proof. 

• Let $x,y\in h\left(R\right)$ and $m+\varphi \left(K\right)\in h(M/\varphi (K\left)\right)$ with $\varphi \left(K\right)\ne xy(m+\varphi (K\left)\right)=xym+\varphi \left(K\right)\in K/\varphi \left(K\right)$. Then $m\in h\left(M\right)$ such that $xym\in K$, but $xym\notin \varphi \left(K\right)$. Hence $xy\in \left(K{:}_{R}M\right)$ or $ym\in K$ or $xm\in K$. So, $xy\in (K/\varphi \left(K\right){:}_{R}M/\varphi \left(K\right))$ or $y(m+\varphi (K\left)\right)\in K/\varphi \left(K\right)$ or $x(m+\varphi (K\left)\right)\in K/\varphi \left(K\right)$, as desired. Conversely, let $x,y\in h\left(R\right)$ and $m\in h\left(M\right)$ such that $xym\in K$ and $xym\notin \varphi \left(K\right)$. Then $m+\varphi \left(K\right)\in h(M/\varphi (K\left)\right)$ with $\varphi \left(K\right)\ne xy(m+\varphi (K\left)\right)\in K/\varphi \left(K\right)$. Hence $xy\in (K/\varphi \left(K\right){:}_{R}M/\varphi \left(K\right))$ or $y(m+\varphi (K\left)\right)\in K/\varphi \left(K\right)$ or $x(m+\varphi (K\left)\right)\in K/\varphi \left(K\right)$. So, $xy\in \left(K{:}_{R}M\right)$ or $ym\in K$ or $xm\in K$. Thus K is a graded $\varphi$-2-abs.submodule.
• Let $x,y\in h\left(R\right)$ and $m+\varphi \left(K\right)\in h(M/\varphi (K\left)\right)$ with $\varphi \left(K\right)\ne xy(m+\varphi (K\left)\right)=xym+\varphi \left(K\right)\in K/\varphi \left(K\right)$. Then $m\in h\left(M\right)$ with $xym\in K$, but $xym\notin \varphi \left(K\right)$. Hence either $xy\in \left(K{:}_{R}M\right)$ or $ym\in Gra{d}_M\left(K\right)$ or $xm\in Gra{d}_M\left(K\right)$. So, $xy\in \left(K{:}_{R}M\right)$ or $y(m+\varphi \left(K\right))\in Gra{d}_M\left(K\right)/\varphi \left(K\right)$ or $x(m+\varphi \left(K\right))\in Gra{d}_M\left(K\right)/\varphi \left(K\right)$. The result holds since $Gra{d}_M\left(K\right)/\varphi \left(K\right)=Gra{d}_{M/\varphi \left(K\right)}(K/\varphi \left(K\right))$. One can easily prove the converse.

One can easily prove (3) and (4) along the same lines. □

Next we consider the behaviour of graded $\varphi$-2-absorbing (primary) submodules under graded homomorphisms. Let M and L be two G-graded R-modules. An R-homomorphism $f:M\to L$ is said to be a graded R-homomorphism if $f\left(M_g\right)\subseteq L_g$ for all $g\in G$ ([2]).

Lemma 2.

([4], Lemma 2.16). Suppose that $f:M\to L$ is a graded R-homomorphism. If N is a graded R-submodule of L, then $f^{-1}\left(N\right)$ is a graded R-submodule of M.

Lemma 3

([17], Lemma 4.8). Suppose that $f:M\to L$ is a graded R-homomorphism. If K is a graded R-submodule of M, then $f\left(K\right)$ is a graded R-submodule of $f\left(M\right)$.

Theorem 4.

Suppose that $f:M\to L$ is a graded R-epimorphism. Let $\varphi :GS\left(M\right)\to GS\left(M\right)\bigcup \{\varnothing \}$ and $\psi :GS\left(L\right)\to GS\left(L\right)\bigcup \{\varnothing \}$ be functions.

1. 

If N is a graded ψ-2-abs. prim. submodule of L and $\varphi \left(f^{-1}\left(N\right)\right)=f^{-1}\left(\psi \left(N\right)\right)$, then $f^{-1}\left(N\right)$ is a graded ϕ-2-abs. prim. of M.

2. 

If K is a graded ϕ-2-abs. prim. submodule of M containing $Ker\left(f\right)$ and $\psi \left(f\right(K\left)\right)=f\left(\varphi \right(K\left)\right)$, then $f\left(K\right)$ is a graded ψ-2-abs. prim. submodule of L.

3. 

If N is a graded ψ-2-abs. submodule of L and $\varphi \left(f^{-1}\left(N\right)\right)=f^{-1}\left(\psi \left(N\right)\right)$, then $f^{-1}\left(N\right)$ is a graded ϕ-2-abs. submodule of M.

4. 

If K is a graded ϕ-2-abs. submodule of M containing $Ker\left(f\right)$ and $\psi \left(f\right(K\left)\right)=f\left(\varphi \right(K\left)\right)$, then $f\left(K\right)$ is a graded ψ-2-abs. submodule of L.

Proof. 

• Since f is epimorphism, $f^{-1}\left(N\right)$ is proper. Let $x,y\in h\left(R\right)$ and $m\in h\left(M\right)$ with $xym\in f^{-1}\left(N\right)$ and $xym\notin f^{-1}\left(\psi \left(N\right)\right)$. Since $xym\in f^{-1}\left(N\right)$, $xyf\left(m\right)\in N$. Also, $\varphi \left(f^{-1}\left(N\right)\right)=f^{-1}\left(\psi \left(N\right)\right)$ implies that $xyf\left(m\right)\notin \psi \left(N\right)$. Thus $xyf\left(m\right)\in N-\psi \left(N\right)$. Then $xy\in \left(N{:}_{R}L\right)$ or $xf\left(m\right)\in Gra{d}_L\left(N\right)$ or $yf\left(m\right)\in Gra{d}_L\left(N\right)$. Thus $xy\in \left(f^{-1}\left(N\right){:}_{R}M\right)$ or $xm\in f^{-1}\left(Gra{d}_L\left(N\right)\right)$ or $ym\in f^{-1}\left(Gra{d}_L\left(N\right)\right)$. Since $f^{-1}\left(Gra{d}_L\left(N\right)\right)\subseteq Gra{d}_M\left(f^{-1}\left(N\right)\right)$ ([18], Theorem 2.16), we should conclude that $f^{-1}\left(N\right)$ is a graded $\varphi$-2-abs. prim. submodule.
• Let $x,y\in h\left(R\right)$ and $s\in h\left(L\right)$ with $xys\in f\left(K\right)-\psi \left(f\right(K\left)\right)$. Because f is a graded epimorphism, there is $m\in h\left(M\right)$ with $s=f\left(m\right)$. Hence, $f\left(xym\right)\in f\left(K\right)$ and so $xym\in K$ as $Ker\left(f\right)\subseteq K$. Because $\psi \left(f\right(K\left)\right)=f\left(\varphi \right(K\left)\right)$, we have $xym\notin \varphi \left(K\right)$. Hence $xym\in K-\varphi \left(K\right)$. It implies that $xy\in \left(K{:}_{R}M\right)$ or $xm\in Gra{d}_M\left(K\right)$ or $ym\in Gra{d}_M\left(K\right)$. Thus $xy\in \left(f\left(K\right){:}_{R}L\right)$ or $xs\in f\left(Gra{d}_M\left(K\right)\right)$ or $ys\in f\left(Gra{d}_M\left(K\right)\right)$. As $Ker\left(f\right)\subseteq K$ implies $f\left(Gra{d}_M\left(K\right)\right)=Gra{d}_L\left(f\left(K\right)\right)$, we are done.

One can easily prove (3) and (4) along the same lines. □

Next we consider the behaviour of graded $\varphi$-2-absorbing (primary) submodules under multiplicative homogeneous set. Let M be a G-graded R-module and $S\subseteq h\left(R\right)$ be a multiplicative set. Then $S^{-1}M$ is a G-graded $S^{-1}R$-module with ${\left(S^{-1}M\right)}_g=\{\frac{m}{s},m\in M_h,s\in S\cap R_{h{g}^{-1}}\}$ for all $g\in G$, and ${\left(S^{-1}R\right)}_g=\left\{\frac{a}{s},a\in R_h,s\in S\cap R_{h{g}^{-1}}\right\}$ for all $g\in G$. If K is a graded R-submodule of M, then $S^{-1}K$ is a graded $S^{-1}R$-submodule of $S^{-1}M$. Let $\varphi :GS\left(M\right)\to GS\left(M\right)\bigcup \{\varnothing \}$ be a function and define ${\varphi }_S:GS\left(S^{-1}M\right)\to GS\left(S^{-1}M\right)\bigcup \{\varnothing \}$ by ${\varphi }_S\left(S^{-1}K\right)=S^{-1}\varphi \left(K\right)$ for $\varphi \left(K\right)\ne \varnothing$ and ${\varphi }_S\left(S^{-1}K\right)=\varnothing$ otherwise, for every graded R-submodule K of M ([10]).

Theorem 5.

Let M be a graded R-module and $S\subseteq h\left(R\right)$ be a multiplicative set.

1. 

If K is a graded ϕ-2-abs. prim. of M and $S^{-1}K\ne S^{-1}M$, then $S^{-1}K$ is a graded ${\varphi }_S$-2-abs. prim. of $S^{-1}M$.

2. 

If K is a graded ϕ-2-abs. of M and $S^{-1}K\ne S^{-1}M$, then $S^{-1}K$ is a graded ${\varphi }_S$-2-abs. of $S^{-1}M$.

Proof. 

1. Let $x/s_1,y/s_2\in h\left(S^{-1}R\right)$ and $m/s\in h\left(S^{-1}M\right)$ with $(x/s_1)(y/s_2)(m/s)\in S^{-1}K-{\varphi }_S\left(S^{-1}K\right)$. Then $x,y\in h\left(R\right)$ and $m\in h\left(M\right)$ with $uxym\in K-\varphi \left(K\right)$ for some $u\in S$, and then $uxm\in Gra{d}_M\left(K\right)$ or $uym\in Gra{d}_M\left(K\right)$ or $xy\in \left(K{:}_{R}M\right)$. So, $(x/s_1)(m/s)=\left(uxm\right)/\left(u{s}_{1}s\right)\in S^{-1}\left(Gra{d}_M\left(K\right)\right)\subseteq Gra{d}_{S^{-1}M}\left(S^{-1}K\right)$ or $(y/s_2)(m/s)=\left(uym\right)/\left(u{s}_{2}s\right)\in Gra{d}_{S^{-1}M}\left(S^{-1}K\right)$ or $(x/s_1)(y/s_2)=\left(xy\right)/\left(s_1{s}_2\right)\in S^{-1}\left(K{:}_{R}M\right)\subseteq \left(S^{-1}K{:}_{S^{-1}R}{S}^{-1}M\right)$.

Similarly, one can easily prove (2). □

Next we consider the behaviour of graded $\varphi$-2-absorbing (primary) submodules under graded Cartesian product. Let $M_1$ be a G-graded $R_1$-module, $M_2$ be a G-graded $R_2$-module and $R=R_1\times R_2$. Then $M=M_1\times M_2$ is a G-graded R-module with $M_g={\left(M_1\right)}_g\times {\left(M_2\right)}_g$ for all $g\in G$, where $R_g={\left(R_1\right)}_g\times {\left(R_2\right)}_g$ for all $g\in G$ ([2]).

Lemma 4

([16], Lemma 3.12). Let $M_1$ be a G-graded $R_1$-module, $M_2$ be a G-graded $R_2$-module, $R=R_1\times R_2$ and $M=M_1\times M_2$. Then $L=N\times K$ is a graded R-submodule of M if and only if N is a graded $R_1$-submodule of $M_1$ and K is a graded $R_2$-submodule of $M_2$.

Lemma 5.

Let $M_1$ be a G-graded $R_1$-module, $M_2$ be a G-graded $R_2$-module, $R=R_1\times R_2$ and $M=M_1\times M_2$. Suppose that ${\psi }_1:GS\left(M_1\right)\to GS\left(M_1\right)\bigcup \{\varnothing \}$, ${\psi }_2:GS\left(M_2\right)\to GS\left(M_2\right)\bigcup \{\varnothing \}$ be functions and $\varphi ={\psi }_1\times {\psi }_2$. Assume that $K=K_1\times M_2$ for some proper graded $R_1$-submodule $K_1$ of $M_1$. If K is a graded ϕ-2-abs. submodule of M, then $K_1$ is a graded ${\psi }_1$-2-abs. submodule of $M_1$.

Proof. 

Let $x_1,y_1\in h\left(R_1\right)$ and $m_1\in h\left(M_1\right)$ with $x_1{y}_1{m}_1\in K_1-{\psi }_1\left(K_1\right)$. Then for $x_2,y_2\in h\left(R_2\right)$ with the same degree as $x_1$, $y_1$ respectively, and for $m_2\in h\left(M_2\right)$ with the same degree as $m_1$, $(x_1,x_2),(y_1,y_2)\in h\left(R\right)$ and $(m_1,m_2)\in h\left(M\right)$ with $(x_1,x_2)(y_1,y_2)(m_1,m_2)\in (K_1\times M_2)-({\psi }_1\left(K_1\right)\times {\psi }_2\left(M_2\right))=K-\varphi \left(K\right)$. Since K is a graded $\varphi$-2-abs. submodule of M, we get either $(x_1,x_2)(y_1,y_2)\in ((K_1\times M_2){:}_R{M}_1\times M_2)$ or $(x_1,x_2)(m_1,m_2)\in (K_1\times M_2)$ or $(y_1,y_2)(m_1,m_2)\in (K_1\times M_2)$. So clearly, we conclude that $x_1{y}_1\in \left(K_1{:}_{R_1}{M}_1\right)$ or $x_1{m}_1\in K_1$ or $y_1{m}_1\in K_1$. Therefore, $K_1$ is a graded ${\psi }_1$-2-abs. submodule of $M_1$. □

Theorem 6.

Let $M_1$ be a G-graded $R_1$-module, $M_2$ be a G-graded $R_2$-module, $R=R_1\times R_2$ and $M=M_1\times M_2$. Suppose that ${\psi }_1:GS\left(M_1\right)\to GS\left(M_1\right)\bigcup \{\varnothing \}$, ${\psi }_2:GS\left(M_2\right)\to GS\left(M_2\right)\bigcup \{\varnothing \}$ be functions and $\varphi ={\psi }_1\times {\psi }_2$. Assume that $K=K_1\times M_2$ for some proper graded $R_1$-submodule $K_1$ of $M_1$.

1. 

If ${\psi }_2\left(M_2\right)=M_2$, then K is a graded ϕ-2-abs. submodule of M if and only if $K_1$ is a graded ${\psi }_1$-2-abs. submodule of $M_1$.

2. 

If ${\psi }_2\left(M_2\right)\ne M_2$, then K is a graded ϕ-2-abs. submodule of M if and only if $K_1$ is a graded 2-abs. submodule of $M_1$.

Proof. 

• Suppose that $K_1$ is a graded ${\psi }_1$-2-abs. submodule of $M_1$. Let $x=(x_1,x_2),y=(y_1,y_2)\in h\left(R\right)$ and $m=(m_1,m_2)\in h\left(M\right)$ with $xym\in K-\varphi \left(K\right)$. Since ${\psi }_2\left(M_2\right)=M_2$, we get that $x_1,y_1\in h\left(R_1\right)$ and $m_1\in h\left(M_1\right)$ with $x_1{y}_1{m}_1\in K_1-{\psi }_1\left(K_1\right)$, and this implies that $x_1{y}_1\in \left(K_1{:}_{R_1}{M}_1\right)$ or $x_1{m}_1\in K_1$ or $y_1{m}_1\in K_1$. Thus either $xy\in \left(K{:}_{R}M\right)$ or $xm\in K$ or $ym\in K$. Hence, K is a graded $\varphi$-2-abs. submodule of M. The converse holds from Lemma 5.
• Suppose that K is a graded $\varphi$-2-abs. submodule of M. Since ${\psi }_2\left(M_2\right)\ne M_2$, there is $m_2\in M_2-{\psi }_2\left(M_2\right)$ and then there is $g\in G$ with ${\left(m_2\right)}_g\in M_2-{\psi }_2\left(M_2\right)$. Suppose that $K_1$ is not a graded 2-abs. of $M_1$. By Lemma 5, $K_1$ is a graded ${\psi }_1$-2-abs. of $M_1$. Hence, there are $x_1,y_1\in h\left(R_1\right)$ and $m_1\in h\left(M_1\right)$ with $x_1{y}_1{m}_1\in {\psi }_1\left(K_1\right)$, $x_1{y}_1\notin \left(K_1{:}_{R_1}{M}_1\right)$, $x_1{m}_1\notin K_1$ and $y_1{m}_1\notin K_1$. So, $(x_1,1)(y_1,1)(m_1,{\left(m_2\right)}_g)\in (K_1\times M_2)-({\psi }_1\left(K_1\right)\times {\psi }_2\left(M_2\right))=(K_1\times M_2)-\varphi (K_1\times M_2)$ which implies that $x_1{y}_1\in \left(K_1{:}_{R_1}{M}_1\right)$ or $x_1{m}_1\in K_1$ or $y_1{m}_1\in K_1$, that is a contradiction. So, $K_1$ is a graded 2-abs. submodule of $M_1$. Conversely, if $K_1$ is a graded 2-abs. submodule of $M_1$, then $K=K_1\times M_2$ is a graded 2-abs. submodule of M by ([6], Theorem 3.3). Hence K is a graded $\varphi$-2-abs. submodule of M for any $\varphi$.

□

Lemma 6.

Let $M_1$ be a G-graded $R_1$-module, $M_2$ be a G-graded $R_2$-module, $R=R_1\times R_2$ and $M=M_1\times M_2$. Suppose that ${\psi }_1:GS\left(M_1\right)\to GS\left(M_1\right)\bigcup \{\varnothing \}$, ${\psi }_2:GS\left(M_2\right)\to GS\left(M_2\right)\bigcup \{\varnothing \}$ be functions and $\varphi ={\psi }_1\times {\psi }_2$. Assume that $K=K_1\times M_2$ for some proper graded $R_1$-submodule $K_1$ of $M_1$. If K is a graded ϕ-2-abs. prim. submodule of M, then $K_1$ is a graded ${\psi }_1$-2-abs. prim. submodule of $M_1$.

Proof. 

The proof uses the same procedure in Lemma 5. □

Theorem 7.

Let $M_1$ be a G-graded $R_1$-module, $M_2$ be a G-graded $R_2$-module, $R=R_1\times R_2$ and $M=M_1\times M_2$. Suppose that ${\psi }_1:GS\left(M_1\right)\to GS\left(M_1\right)\bigcup \{\varnothing \}$, ${\psi }_2:GS\left(M_2\right)\to GS\left(M_2\right)\bigcup \{\varnothing \}$ be functions and $\varphi ={\psi }_1\times {\psi }_2$. Assume that $K=K_1\times M_2$ for some proper graded $R_1$-submodule $K_1$ of $M_1$.

1. 

If ${\psi }_2\left(M_2\right)=M_2$, then K is a graded ϕ-2-abs. prim. submodule of M if and only if $K_1$ is a graded ${\psi }_1$-2-abs. prim. submodule of $M_1$.

2. 

If ${\psi }_2\left(M_2\right)\ne M_2$, then K is a graded ϕ-2-abs. prim. submodule of M if and only if $K_1$ is a graded 2-abs. prim. submodule of $M_1$.

Proof. 

• It can be easily proved by using a similar procedure in Theorem 6 (1).
• Suppose that $K_1$ is a graded 2-abs. prim. submodule of $M_1$. Then $K=K_1\times M_2$ is a graded 2-abs. prim. submodule of M by ([7], Theorem 18). Hence K is a graded $\varphi$-2-abs. submodule of M for any $\varphi$. The remainder of the proof is similar to that of Theorem 6 (2).

□

3. More Results

In this section, we introduce more results concerning the g-components of M, $g\in G$.

Theorem 8.

Let M be a G-graded R-module $g\in G$ and K be a g-ϕ-primary submodule of M. Suppose that $x\in R_e$ and $m\in M_g$ such that $xm\in \varphi \left(K\right)$, $x\notin Grad\left(\left(K{:}_{R}M\right)\right)$ and $m\notin K$. Then

1. 

$x{K}_g\subseteq \varphi \left(K\right)$.

2. 

$\left(K{:}_{R_e}M\right)m\subseteq \varphi \left(K\right)$.

3. 

$\left(K{:}_{R_e}M\right)K_g\subseteq \varphi \left(K\right)$.

Proof. 

• Suppose that $x{K}_g⊈\varphi \left(K\right)$. Then there is $k\in K_g$ with $xk\notin \varphi \left(K\right)$, and then $x(m+k)\notin \varphi \left(K\right)$. Since $x(m+k)\in K$ and $x\notin Grad\left(\left(K{:}_{R}M\right)\right)$, we deduce that $m+k\in K$ as K is a g-$\varphi$-primary R-submodule of M. So $m\in K$, which is a contradiction. So, $x{K}_g\subseteq \varphi \left(K\right)$.
• Suppose that $\left(K{:}_{R_e}M\right)m⊈\varphi \left(K\right)$. Then there is $y\in \left(K{:}_{R_e}M\right)$ with $ym\notin \varphi \left(K\right)$, and then $(x+y)m\notin \varphi \left(K\right)$ as $xm\in \varphi \left(K\right)$. Since $ym\in K$, we get $(x+y)m\in K$. Since $m\notin K$, so $x+y\in Grad\left(\left(K{:}_{R}M\right)\right)$. Hence $x\in Grad\left(\left(K{:}_{R}M\right)\right)$, which is a contradiction.
• Suppose that $\left(K{:}_{R_e}M\right)K_g⊈\varphi \left(K\right)$. Then there exist $y\in \left(K{:}_{R_e}M\right)$ and $k\in K_g$ such that $yk\notin \varphi \left(K\right)$. By (1) and (2), $(x+y)(m+k)\in K-\varphi \left(K\right)$. So, either $x+y\in Grad\left(\left(K{:}_{R}M\right)\right)$ or $m+k\in K$. Thus we have either $x\in Grad\left(\left(K{:}_{R}M\right)\right)$ or $m\in K$, which is a contradiction.

□

Remark 3.

Note that if K is a g-ϕ-primary submodule of M which is not g-primary, then there are $x\in R_e$ and $m\in M_g$ with $xm\in \varphi \left(K\right)$, $x\notin Grad\left(\left(K{:}_{R}M\right)\right)$ and $m\notin K$. So, every g-ϕ-primary submodule, which is not g-primary, satisfies the assumptions of Theorem 8.

Theorem 9.

Let M be a G-graded R-module, $g\in G$ and K be a g-ϕ-2-abs. submodule of M. Suppose that $x,y\in R_e$ and $m\in M_g$ with $xym\in \varphi \left(K\right)$, $xy\notin \left(K{:}_{R}M\right)$, $xm\notin K$ and $ym\notin K$. Then

1. 

$xy{K}_g\subseteq \varphi \left(K\right)$.

2. 

$x\left(K{:}_{R_e}M\right)m\subseteq \varphi \left(K\right)$.

3. 

$y\left(K{:}_{R_e}M\right)m\subseteq \varphi \left(K\right)$.

4. 

${\left(K{:}_{R_e}M\right)}^{2}m\subseteq \varphi \left(K\right)$.

Proof. 

• Suppose that $xy{K}_g⊈\varphi \left(K\right)$. Then there is $k\in K_g$ with $xyk\notin \varphi \left(K\right)$, and then $xy(m+k)\notin \varphi \left(K\right)$. Since $xy(m+k)=xym+xyk\in K$ and $xy\notin \left(K{:}_{R}M\right)$, we conclude that $x(m+k)\in K$ or $y(m+k)\in K$. So, $xm\in K$ or $ym\in K$, which is a contradiction. Thus $xyK\subseteq \varphi \left(K\right)$.
• Suppose that $x\left(K{:}_{R_e}M\right)m⊈\varphi \left(K\right)$. Then there is $a\in \left(K{:}_{R_e}M\right)$ with $xam\notin \varphi \left(K\right)$, and then $x(y+a)m\notin \varphi \left(K\right)$ as $xym\in \varphi \left(K\right)$. Since $am\in K$, we obtain that $x(y+a)m\in K$. Then $xm\in K$ or $(y+a)m\in K$ or $x(y+a)\in \left(K{:}_{R}M\right)$. Hence $xm\in K$ or $ym\in K$ or $xy\in \left(K{:}_{R}M\right)$, which is a contradiction. Hence, $x\left(K{:}_{R_e}M\right)m\subseteq \varphi \left(K\right)$.
• It can be easily proved by using a similar procedure in part (2).
• Assume that ${\left(K{:}_{R_e}M\right)}^{2}m⊈\varphi \left(K\right)$. Then there exist $a,b\in \left(K{:}_{R_e}M\right)$ with $abm\notin \varphi \left(K\right)$, and then by parts (2) and (3), $(x+a)(y+b)m\notin \varphi \left(K\right)$. Clearly, $(x+a)(y+b)m\in K$. Then $(x+a)m\in K$ or $(y+b)m\in K$ or $(x+a)(y+b)\in \left(K{:}_{R}M\right)$. Therefore, $xm\in K$ or $ym\in K$ or $xy\in \left(K{:}_{R}M\right)$, which is a contradiction. Consequently, ${\left(K{:}_{R_e}M\right)}^{2}m\subseteq \varphi \left(K\right)$.

□

Remark 4.

Note that if K is a g-ϕ-2-abs. submodule of M which is not g-2-abs., then there are $x,y\in R_e$ and $m\in M_g$ with $xym\in \varphi \left(K\right)$, $xy\notin \left(K{:}_{R}M\right)$, $xm\notin K$ and $ym\notin K$. So, every g-ϕ-2-abs. submodule, which is not g-2-abs., satisfies the assumptions of Theorem 9.

Theorem 10.

Let M be a G-graded R-module, $g\in G$ and K be a g-ϕ-2-abs. prim. submodule of M. Consider $x,y\in R_e$ and $m\in M_g$ with $xym\in \varphi \left(K\right)$, $xy\notin \left(K{:}_{R}M\right)$, $xm\notin Gra{d}_M\left(K\right)$ and $ym\notin Gra{d}_M\left(K\right)$. Then

1. 

$xy{K}_g\subseteq \varphi \left(K\right)$.

2. 

$x\left(K{:}_{R_e}M\right)m\subseteq \varphi \left(K\right)$.

3. 

$y\left(K{:}_{R_e}M\right)m\subseteq \varphi \left(K\right)$.

4. 

${\left(K{:}_{R_e}M\right)}^{2}m\subseteq \varphi \left(K\right)$.

Proof. 

This can be easily proved in a similar way as Theorem 9. □

Remark 5.

Note that if K is a g-ϕ-2-abs. prim. submodule of M which is not g-2-abs. prim., then there are $x,y\in R_e$ and $m\in M_g$ with $xym\in \varphi \left(K\right)$, $xy\notin \left(K{:}_{R}M\right)$, $xm\notin Gra{d}_M\left(K\right)$ and $ym\notin Gra{d}_M\left(K\right)$. So, every g-ϕ-2-abs. prim. submodule, which is not g-2-abs. prim., satisfies the assumptions of Theorem 10.

Theorem 11.

Let M be a G-graded R-module and $g\in G$. If K is a g-ϕ-2-abs. prim. submodule of M that is not g-2-abs. prim., then ${\left(K{:}_{R_e}M\right)}^2{K}_g\subseteq \varphi \left(K\right)$.

Proof. 

Since K is a g-$\varphi$-2-abs. prim. submodule of M that is not g-2-abs. prim., there are $x,y\in R_e$ and $m\in M_g$ such that $xym\in \varphi \left(K\right)$, $xy\notin \left(K{:}_{R}M\right)$, $xm\notin Gra{d}_M\left(K\right)$ and $ym\notin Gra{d}_M\left(K\right)$. Suppose that ${\left(K{:}_{R_e}M\right)}^2{K}_g⊈\varphi \left(K\right)$. Then there are $a,b\in \left(K{:}_{R_e}M\right)$ and $k\in K_g$ such that $abk\notin \varphi \left(K\right)$. By Theorem 10, we get $(x+a)(y+b)(m+k)\in K-\varphi \left(K\right)$. So, $(x+a)(m+k)\in Gra{d}_M\left(K\right)$ or $(y+b)(m+k)\in Gra{d}_M\left(K\right)$ or $(x+a)(y+b)\in \left(K{:}_{R}M\right)$. Therefore, $xm\in Gra{d}_M\left(K\right)$ or $ym\in Gra{d}_M\left(K\right)$ or $xy\in \left(K{:}_{R}M\right)$, which is a contradiction. Hence, ${\left(K{:}_{R_e}M\right)}^2{K}_g\subseteq \varphi \left(K\right)$. □

4. Conclusions

In this study, we introduced the notions of graded $\varphi$-2-abs. and graded $\varphi$-2-abs. prim. submodules as generalizations of the notions of graded 2-abs. and graded 2-abs. prim. submodules. Several properties of these new types of graded submodules have been given. As a proposal of further work on the topic, we are going to study the concept of graded $\varphi$-r-ideals as a generalization of the concept of graded r-ideals.