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Article

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

by
Azzh Saad Alshehry
1,
Malik Bataineh
2 and
Rashid Abu-Dawwas
3,*
1
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2
Department of Mathematics, Jordan University of Science and Technology, Irbid 22110, Jordan
3
Department of Mathematics, Yarmouk University, Irbid 21163, Jordan
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(10), 1083; https://doi.org/10.3390/math9101083
Submission received: 15 March 2021 / Revised: 24 April 2021 / Accepted: 28 April 2021 / Published: 11 May 2021
(This article belongs to the Section A: Algebra and Logic)

Abstract

:
The main goal of this article is to explore the concepts of graded ϕ -2-absorbing and graded ϕ -2-absorbing primary submodules as a new generalization of the concepts of graded 2-absorbing and graded 2-absorbing primary submodules. Let ϕ : G S ( M ) G S ( M ) { } be a function, where G S ( M ) denotes the collection of graded R-submodules of M. A proper K G S ( M ) is said to be a graded ϕ -2-absorbing R-submodule of M if whenever x , y are homogeneous elements of R and s is a homogeneous element of M with x y s K ϕ ( K ) , then x s K or y s K or x y ( K : R M ) , and we call K a graded ϕ -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 x y s K ϕ ( K ) , then x s or y s is in the graded radical of K or x y ( K : R M ) . 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 R = g G R g with R g R h R g h for all g , h G where R g is an additive subgroup of R for all g G . The elements of R g are called homogeneous of degree g. If x R , then x can be written uniquely as g 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 R e . The set of all homogeneous elements of R is h ( R ) = g G R g . Assume that M is a left unitary R-module. Then M is said to be G-graded if M = g G M g with R g M h M g h for all g , h G where M g is an additive subgroup of M for all g 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 G . If x M , then x can be written uniquely as g G x g , where x g is the component of x in M g . The set of all homogeneous elements of M is h ( M ) = g 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 K = g G ( K M g ) , i.e., for x K , x = g G x g where x g K for all g G . An R-submodule of a graded R-module need not be graded. For more details and terminology, see [1,2].
Specifically, A n n R ( M ) = ( 0 : R M ) 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 G r a d ( I ) , and is defined to be the collection of all r R so that for every g G , there exists n g N for which r g n g I . One can see that if r h ( R ) , then r G r a d ( I ) if and only if r n I for some n N . In fact, G r a d ( I ) 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 G r a d M ( K ) 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 G r a d M ( K ) = M .
A graded prime (resp. graded primary) R-submodule is a graded R-submodule K M with the property that for a h ( R ) and s h ( M ) such that a s K implies that s K or a ( K : R M ) (resp. s K or a G r a d ( ( K : R M ) ) ) . 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 M is called graded weakly prime if whenever a h ( R ) and s h ( M ) with 0 a s K , then s K or a ( K : R M ) . The concept of graded ϕ -prime submodules has been introduced in [10]. Let ϕ : G S ( M ) G S ( M ) { } be a function, where G S ( M ) denotes the collection of graded submodules of M. A graded R-submodule K M is called graded ϕ -prime if whenever a h ( R ) and s h ( M ) such that a s K ϕ ( K ) , then s K or a ( K : R M ) .
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 h ( R ) and x y z I (resp. 0 x y z I ), then x y I or x z I or y z 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 h ( R ) with x y z I , then x y I or x z G r a d ( I ) or y z G r a d ( I ) . 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 h ( R ) and m h ( M ) with x y m K (resp. 0 x y m K ), then x y ( K : R M ) or x m G r a d M ( K ) or y m G r a d M ( K ) .
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 = I M . In this instance, I = ( K : R M ) . 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 = I M and K = J M for some graded ideals I and J of R. The product of N and K is denoted by N K is defined by N K = I J M . Then the product of N and K is independent of presentations of N and K. In fact, as I J is a graded ideal of R (see [2]), N K is a graded R-submodule of M and N K N K . Moreover, for x , y h ( M ) , by x y , we mean the product of R x and R y . 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 G r a d M ( N ) = G r a d ( ( N : R M ) ) M .
In this article, our aim is to extend the notion of graded 2-absorbing submodules to graded ϕ -2-absorbing submodules using similar techniques to that are used in [10], and also to extend graded 2-absorbing primary submodules to graded ϕ -2-absorbing primary submodules. Our study is inspired by [15].

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

This segment includes proposing and examining the notions of graded ϕ -2-absorbing and graded ϕ -2-absorbing primary submodules.
Definition 1.
Let M be a G-graded R-module and ϕ : G S ( M ) G S ( M ) { } be a function.
1. 
A graded R-submodule K M is called graded ϕ-primary submodule if whenever a h ( R ) and s h ( M ) with a s K ϕ ( K ) , then s K or a G r a d ( ( K : R M ) ) .
2. 
A graded R-submodule K with K g M g for some g G is called g-ϕ-primary submodule if whenever a R e and s M g with a s K ϕ ( K ) , then s K or a G r a d ( ( K : R M ) ) .
3. 
A graded R-submodule K M is called graded ϕ-2-absorbing submodule (ϕ-2-abs. submodule) if whenever x , y h ( R ) and s h ( M ) with x y s K ϕ ( K ) , then x s K or y s K or x y ( K : R M ) .
4. 
A graded R-submodule K with K g M g for some g G is called g-ϕ-2-absorbing submodule (g-ϕ-2-abs. submodule) if whenever x , y R e and s M g with x y s K ϕ ( K ) , then x s K or y s K or x y ( K : R M ) .
5. 
A graded R-submodule K M of M is called graded ϕ-2-absorbing primary submodule (ϕ-2-abs. prim. submodule) if whenever x , y h ( R ) and s h ( M ) with x y s K ϕ ( K ) , then x s G r a d M ( K ) or y s G r a d M ( K ) or x y ( K : R M ) .
6. 
A graded R-submodule K with K g M g for some g G is called g-ϕ-2-absorbing primary submodule (g-ϕ-2-abs. prim. submodule) if whenever x , y R e and s M g with x y s K ϕ ( K ) , then x s G r a d M ( K ) or y s G r a d M ( K ) or x y ( K : R M ) .
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 h ( R ) and m h ( M ) such that r m K ϕ ( K ) , then either m K or r G r a d M ( K ) .
Remark 1.
1. 
For a graded ϕ-primary R-submodule K of a graded multiplication R-module M, we have the following functions:
ϕ ( K ) = graded primary submodule,
ϕ 0 ( K ) = { 0 } graded weakly primary submodule,
ϕ 2 ( K ) = K 2 graded almost primary submodule,
ϕ n ( K ) = K n graded n-almost primary submodule, and
ϕ ω ( K ) = n = 1 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
ϕ ( K ) = graded 2-abs. submodule (resp. graded 2-abs. prim. submodule),
ϕ 0 ( K ) = { 0 } graded weakly 2-abs. submodule (resp. graded weakly 2-abs. prim. submodule),
ϕ 2 ( K ) = K 2 graded almost 2-abs. submodule (resp. graded almost 2-abs. prim. submodule),
ϕ n ( K ) = K n graded n-almost 2-abs. submodule (resp. graded n-almost 2-abs. prim. submodule), and
ϕ ω ( K ) = n = 1 K n graded ω-2-abs. submodule (resp. graded ω-2-abs. prim. submodule).
3. 
For functions ϕ , ψ : G S ( M ) G S ( M ) { } , we write ϕ ψ if ϕ ( K ) ψ ( K ) for all K G S ( M ) . Obviously, therefore, we have the next order:
ψ ψ 0 ψ ω . . . ψ n + 1 ψ n . . . ψ 2 ψ 1 .
4. 
If ϕ ψ , 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 ϕ ( K ) = K ( K ϕ ( K ) ) for any graded R-submodule K, we may assume ϕ ( K ) 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 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 2 ⇒K is graded almost 2-abs. prim. submodule.
5. 
Suppose that G r a d M ( K ) = 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 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 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 h ( R ) and m h ( M ) with x y m K ϕ ( K ) . suppose that y m G r a d M ( K ) . Then y m K and then x G r a d ( ( K : R M ) ) as K is a graded ϕ -primary R-submodule. Therefore, x m G r a d ( ( K : R M ) M = G r a d M ( K ) . Consequently, K is graded ϕ -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 ϕ ω ( K ) = n = 1 K n = K . Thus K is a graded ω -2-abs.. By (3), we conclude that K is a graded n-almost 2-abs. for all n 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 2 . Let x , y h ( R ) and m h ( M ) with x y m K but x y m n = 1 K n . Hence x y m K n for some n 2 . Since K is graded n-almost 2-abs. (resp. graded n-almost 2-abs. prim.) for all n 2 , this implies either x y ( K : R M ) or y m K or x m K (resp. x y ( K : R M ) or y m G r a d M ( K ) or x m G r a d M ( K ) ). 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 ϕ ( K ) is a graded 2-abs. prim. submodule of M, then K is a graded-2-abs. prim. submodule of M.
Proof. 
Let x , y h ( R ) and m h ( M ) such that x y m K . If x y m ϕ ( K ) , then we conclude that x m G r a d M ( ϕ ( K ) ) G r a d M ( K ) or y m G r a d M ( ϕ ( K ) ) G r a d M ( K ) or x y ( ϕ ( K ) : R M ) ( K : R M ) since ϕ ( K ) is graded 2-abs. prim., and so the result holds. If x y m ϕ ( K ) , then the result holds easily since K is graded ϕ -2-abs. prim. submodule. □
Theorem 2.
Let M be a graded R-module and K be a graded R-submodule of M with ϕ ( G r a d M ( K ) ) ϕ ( K ) . If G r a d M ( K ) is a graded ϕ-prime submodule of M, then K is a graded ϕ-2-abs. prim. submodule of M.
Proof. 
Let x , y h ( R ) and m h ( M ) with x y m K ϕ ( K ) and x m G r a d M ( K ) . Because G r a d M ( K ) is a graded ϕ -prime and x y m G r a d M ( K ) ϕ ( G r a d M ( K ) ) , y ( G r a d M ( K ) : R M ) . So, y m G r a d M ( K ) . Consequently, K is a graded ϕ -2-abs. prim. of M. □
Next we consider the behaviour of graded ϕ -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 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 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 / ϕ ( K ) is a graded weakly 2-abs. submodule of M / ϕ ( K ) .
2. 
K is a graded ϕ-2-abs. prim. submodule of M if and only if K / ϕ ( K ) is a graded weakly 2-abs. prim. submodule of M / ϕ ( K ) .
3. 
K is a graded ϕ-prime submodule of M if and only if K / ϕ ( K ) is a graded weakly prime submodule of M / ϕ ( K ) .
4. 
K is a graded ϕ-primary submodule of M if and only if K / ϕ ( K ) is a graded weakly primary submodule of M / ϕ ( K ) .
Proof. 
  • Let x , y h ( R ) and m + ϕ ( K ) h ( M / ϕ ( K ) ) with ϕ ( K ) x y ( m + ϕ ( K ) ) = x y m + ϕ ( K ) K / ϕ ( K ) . Then m h ( M ) such that x y m K , but x y m ϕ ( K ) . Hence x y ( K : R M ) or y m K or x m K . So, x y ( K / ϕ ( K ) : R M / ϕ ( K ) ) or y ( m + ϕ ( K ) ) K / ϕ ( K ) or x ( m + ϕ ( K ) ) K / ϕ ( K ) , as desired. Conversely, let x , y h ( R ) and m h ( M ) such that x y m K and x y m ϕ ( K ) . Then m + ϕ ( K ) h ( M / ϕ ( K ) ) with ϕ ( K ) x y ( m + ϕ ( K ) ) K / ϕ ( K ) . Hence x y ( K / ϕ ( K ) : R M / ϕ ( K ) ) or y ( m + ϕ ( K ) ) K / ϕ ( K ) or x ( m + ϕ ( K ) ) K / ϕ ( K ) . So, x y ( K : R M ) or y m K or x m K . Thus K is a graded ϕ -2-abs.submodule.
  • Let x , y h ( R ) and m + ϕ ( K ) h ( M / ϕ ( K ) ) with ϕ ( K ) x y ( m + ϕ ( K ) ) = x y m + ϕ ( K ) K / ϕ ( K ) . Then m h ( M ) with x y m K , but x y m ϕ ( K ) . Hence either x y ( K : R M ) or y m G r a d M ( K ) or x m G r a d M ( K ) . So, x y ( K : R M ) or y ( m + ϕ ( K ) ) G r a d M ( K ) / ϕ ( K ) or x ( m + ϕ ( K ) ) G r a d M ( K ) / ϕ ( K ) . The result holds since G r a d M ( K ) / ϕ ( K ) = G r a d M / ϕ ( K ) ( K / ϕ ( K ) ) . One can easily prove the converse.
One can easily prove (3) and (4) along the same lines. □
Next we consider the behaviour of graded ϕ -2-absorbing (primary) submodules under graded homomorphisms. Let M and L be two G-graded R-modules. An R-homomorphism f : M L is said to be a graded R-homomorphism if f ( M g ) L g for all g G ([2]).
Lemma 2.
([4], Lemma 2.16). Suppose that f : M L is a graded R-homomorphism. If N is a graded R-submodule of L, then f 1 ( N ) is a graded R-submodule of M.
Lemma 3
([17], Lemma 4.8). Suppose that f : M L is a graded R-homomorphism. If K is a graded R-submodule of M, then f ( K ) is a graded R-submodule of f ( M ) .
Theorem 4.
Suppose that f : M L is a graded R-epimorphism. Let ϕ : G S ( M ) G S ( M ) { } and ψ : G S ( L ) G S ( L ) { } be functions.
1. 
If N is a graded ψ-2-abs. prim. submodule of L and ϕ ( f 1 ( N ) ) = f 1 ( ψ ( N ) ) , then f 1 ( N ) is a graded ϕ-2-abs. prim. of M.
2. 
If K is a graded ϕ-2-abs. prim. submodule of M containing K e r ( f ) and ψ ( f ( K ) ) = f ( ϕ ( K ) ) , then f ( K ) is a graded ψ-2-abs. prim. submodule of L.
3. 
If N is a graded ψ-2-abs. submodule of L and ϕ ( f 1 ( N ) ) = f 1 ( ψ ( N ) ) , then f 1 ( N ) is a graded ϕ-2-abs. submodule of M.
4. 
If K is a graded ϕ-2-abs. submodule of M containing K e r ( f ) and ψ ( f ( K ) ) = f ( ϕ ( K ) ) , then f ( K ) is a graded ψ-2-abs. submodule of L.
Proof. 
  • Since f is epimorphism, f 1 ( N ) is proper. Let x , y h ( R ) and m h ( M ) with x y m f 1 ( N ) and x y m f 1 ( ψ ( N ) ) . Since x y m f 1 ( N ) , x y f ( m ) N . Also, ϕ ( f 1 ( N ) ) = f 1 ( ψ ( N ) ) implies that x y f ( m ) ψ ( N ) . Thus x y f ( m ) N ψ ( N ) . Then x y ( N : R L ) or x f ( m ) G r a d L ( N ) or y f ( m ) G r a d L ( N ) . Thus x y ( f 1 ( N ) : R M ) or x m f 1 ( G r a d L ( N ) ) or y m f 1 ( G r a d L ( N ) ) . Since f 1 ( G r a d L ( N ) ) G r a d M ( f 1 ( N ) ) ([18], Theorem 2.16), we should conclude that f 1 ( N ) is a graded ϕ -2-abs. prim. submodule.
  • Let x , y h ( R ) and s h ( L ) with x y s f ( K ) ψ ( f ( K ) ) . Because f is a graded epimorphism, there is m h ( M ) with s = f ( m ) . Hence, f ( x y m ) f ( K ) and so x y m K as K e r ( f ) K . Because ψ ( f ( K ) ) = f ( ϕ ( K ) ) , we have x y m ϕ ( K ) . Hence x y m K ϕ ( K ) . It implies that x y ( K : R M ) or x m G r a d M ( K ) or y m G r a d M ( K ) . Thus x y ( f ( K ) : R L ) or x s f ( G r a d M ( K ) ) or y s f ( G r a d M ( K ) ) . As K e r ( f ) K implies f ( G r a d M ( K ) ) = G r a d L ( f ( K ) ) , we are done.
One can easily prove (3) and (4) along the same lines. □
Next we consider the behaviour of graded ϕ -2-absorbing (primary) submodules under multiplicative homogeneous set. Let M be a G-graded R-module and S h ( R ) be a multiplicative set. Then S 1 M is a G-graded S 1 R -module with ( S 1 M ) g = { m s , m M h , s S R h g 1 } for all g G , and ( S 1 R ) g = a s , a R h , s S R h g 1 for all g 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 ϕ : G S ( M ) G S ( M ) { } be a function and define ϕ S : G S ( S 1 M ) G S ( S 1 M ) { } by ϕ S ( S 1 K ) = S 1 ϕ ( K ) for ϕ ( K ) and ϕ S ( S 1 K ) = otherwise, for every graded R-submodule K of M ([10]).
Theorem 5.
Let M be a graded R-module and S h ( R ) be a multiplicative set.
1. 
If K is a graded ϕ-2-abs. prim. of M and S 1 K S 1 M , then S 1 K is a graded ϕ S -2-abs. prim. of S 1 M .
2. 
If K is a graded ϕ-2-abs. of M and S 1 K S 1 M , then S 1 K is a graded ϕ S -2-abs. of S 1 M .
Proof. 
1. Let x / s 1 , y / s 2 h ( S 1 R ) and m / s h ( S 1 M ) with ( x / s 1 ) ( y / s 2 ) ( m / s ) S 1 K ϕ S ( S 1 K ) . Then x , y h ( R ) and m h ( M ) with u x y m K ϕ ( K ) for some u S , and then u x m G r a d M ( K ) or u y m G r a d M ( K ) or x y ( K : R M ) . So, ( x / s 1 ) ( m / s ) = ( u x m ) / ( u s 1 s ) S 1 ( G r a d M ( K ) ) G r a d S 1 M ( S 1 K ) or ( y / s 2 ) ( m / s ) = ( u y m ) / ( u s 2 s ) G r a d S 1 M ( S 1 K ) or ( x / s 1 ) ( y / s 2 ) = ( x y ) / ( s 1 s 2 ) S 1 ( K : R M ) ( S 1 K : S 1 R S 1 M ) .
Similarly, one can easily prove (2). □
Next we consider the behaviour of graded ϕ -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 × R 2 . Then M = M 1 × M 2 is a G-graded R-module with M g = ( M 1 ) g × ( M 2 ) g for all g G , where R g = ( R 1 ) g × ( R 2 ) g for all g 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 × R 2 and M = M 1 × M 2 . Then L = N × 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 × R 2 and M = M 1 × M 2 . Suppose that ψ 1 : G S ( M 1 ) G S ( M 1 ) { } , ψ 2 : G S ( M 2 ) G S ( M 2 ) { } be functions and ϕ = ψ 1 × ψ 2 . Assume that K = K 1 × 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 ψ 1 -2-abs. submodule of M 1 .
Proof. 
Let x 1 , y 1 h ( R 1 ) and m 1 h ( M 1 ) with x 1 y 1 m 1 K 1 ψ 1 ( K 1 ) . Then for x 2 , y 2 h ( R 2 ) with the same degree as x 1 , y 1 respectively, and for m 2 h ( M 2 ) with the same degree as m 1 , ( x 1 , x 2 ) , ( y 1 , y 2 ) h ( R ) and ( m 1 , m 2 ) h ( M ) with ( x 1 , x 2 ) ( y 1 , y 2 ) ( m 1 , m 2 ) ( K 1 × M 2 ) ( ψ 1 ( K 1 ) × ψ 2 ( M 2 ) ) = K ϕ ( K ) . Since K is a graded ϕ -2-abs. submodule of M, we get either ( x 1 , x 2 ) ( y 1 , y 2 ) ( ( K 1 × M 2 ) : R M 1 × M 2 ) or ( x 1 , x 2 ) ( m 1 , m 2 ) ( K 1 × M 2 ) or ( y 1 , y 2 ) ( m 1 , m 2 ) ( K 1 × M 2 ) . So clearly, we conclude that x 1 y 1 ( K 1 : R 1 M 1 ) or x 1 m 1 K 1 or y 1 m 1 K 1 . Therefore, K 1 is a graded ψ 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 × R 2 and M = M 1 × M 2 . Suppose that ψ 1 : G S ( M 1 ) G S ( M 1 ) { } , ψ 2 : G S ( M 2 ) G S ( M 2 ) { } be functions and ϕ = ψ 1 × ψ 2 . Assume that K = K 1 × M 2 for some proper graded R 1 -submodule K 1 of M 1 .
1. 
If ψ 2 ( M 2 ) = M 2 , then K is a graded ϕ-2-abs. submodule of M if and only if K 1 is a graded ψ 1 -2-abs. submodule of M 1 .
2. 
If ψ 2 ( M 2 ) 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 ψ 1 -2-abs. submodule of M 1 . Let x = ( x 1 , x 2 ) , y = ( y 1 , y 2 ) h ( R ) and m = ( m 1 , m 2 ) h ( M ) with x y m K ϕ ( K ) . Since ψ 2 ( M 2 ) = M 2 , we get that x 1 , y 1 h ( R 1 ) and m 1 h ( M 1 ) with x 1 y 1 m 1 K 1 ψ 1 ( K 1 ) , and this implies that x 1 y 1 ( K 1 : R 1 M 1 ) or x 1 m 1 K 1 or y 1 m 1 K 1 . Thus either x y ( K : R M ) or x m K or y m K . Hence, K is a graded ϕ -2-abs. submodule of M. The converse holds from Lemma 5.
  • Suppose that K is a graded ϕ -2-abs. submodule of M. Since ψ 2 ( M 2 ) M 2 , there is m 2 M 2 ψ 2 ( M 2 ) and then there is g G with ( m 2 ) g M 2 ψ 2 ( M 2 ) . Suppose that K 1 is not a graded 2-abs. of M 1 . By Lemma 5, K 1 is a graded ψ 1 -2-abs. of M 1 . Hence, there are x 1 , y 1 h ( R 1 ) and m 1 h ( M 1 ) with x 1 y 1 m 1 ψ 1 ( K 1 ) , x 1 y 1 ( K 1 : R 1 M 1 ) , x 1 m 1 K 1 and y 1 m 1 K 1 . So, ( x 1 , 1 ) ( y 1 , 1 ) ( m 1 , ( m 2 ) g ) ( K 1 × M 2 ) ( ψ 1 ( K 1 ) × ψ 2 ( M 2 ) ) = ( K 1 × M 2 ) ϕ ( K 1 × M 2 ) which implies that x 1 y 1 ( K 1 : R 1 M 1 ) or x 1 m 1 K 1 or y 1 m 1 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 × M 2 is a graded 2-abs. submodule of M by ([6], Theorem 3.3). Hence K is a graded ϕ -2-abs. submodule of M for any ϕ .
Lemma 6.
Let M 1 be a G-graded R 1 -module, M 2 be a G-graded R 2 -module, R = R 1 × R 2 and M = M 1 × M 2 . Suppose that ψ 1 : G S ( M 1 ) G S ( M 1 ) { } , ψ 2 : G S ( M 2 ) G S ( M 2 ) { } be functions and ϕ = ψ 1 × ψ 2 . Assume that K = K 1 × 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 ψ 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 × R 2 and M = M 1 × M 2 . Suppose that ψ 1 : G S ( M 1 ) G S ( M 1 ) { } , ψ 2 : G S ( M 2 ) G S ( M 2 ) { } be functions and ϕ = ψ 1 × ψ 2 . Assume that K = K 1 × M 2 for some proper graded R 1 -submodule K 1 of M 1 .
1. 
If ψ 2 ( M 2 ) = M 2 , then K is a graded ϕ-2-abs. prim. submodule of M if and only if K 1 is a graded ψ 1 -2-abs. prim. submodule of M 1 .
2. 
If ψ 2 ( M 2 ) 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 × M 2 is a graded 2-abs. prim. submodule of M by ([7], Theorem 18). Hence K is a graded ϕ -2-abs. submodule of M for any ϕ . 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 G .
Theorem 8.
Let M be a G-graded R-module g G and K be a g-ϕ-primary submodule of M. Suppose that x R e and m M g such that x m ϕ ( K ) , x G r a d ( ( K : R M ) ) and m K . Then
1. 
x K g ϕ ( K ) .
2. 
( K : R e M ) m ϕ ( K ) .
3. 
( K : R e M ) K g ϕ ( K ) .
Proof. 
  • Suppose that x K g ϕ ( K ) . Then there is k K g with x k ϕ ( K ) , and then x ( m + k ) ϕ ( K ) . Since x ( m + k ) K and x G r a d ( ( K : R M ) ) , we deduce that m + k K as K is a g- ϕ -primary R-submodule of M. So m K , which is a contradiction. So, x K g ϕ ( K ) .
  • Suppose that ( K : R e M ) m ϕ ( K ) . Then there is y ( K : R e M ) with y m ϕ ( K ) , and then ( x + y ) m ϕ ( K ) as x m ϕ ( K ) . Since y m K , we get ( x + y ) m K . Since m K , so x + y G r a d ( ( K : R M ) ) . Hence x G r a d ( ( K : R M ) ) , which is a contradiction.
  • Suppose that ( K : R e M ) K g ϕ ( K ) . Then there exist y ( K : R e M ) and k K g such that y k ϕ ( K ) . By (1) and (2), ( x + y ) ( m + k ) K ϕ ( K ) . So, either x + y G r a d ( ( K : R M ) ) or m + k K . Thus we have either x G r a d ( ( K : R M ) ) or m 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 R e and m M g with x m ϕ ( K ) , x G r a d ( ( K : R M ) ) and m 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 G and K be a g-ϕ-2-abs. submodule of M. Suppose that x , y R e and m M g with x y m ϕ ( K ) , x y ( K : R M ) , x m K and y m K . Then
1. 
x y K g ϕ ( K ) .
2. 
x ( K : R e M ) m ϕ ( K ) .
3. 
y ( K : R e M ) m ϕ ( K ) .
4. 
( K : R e M ) 2 m ϕ ( K ) .
Proof. 
  • Suppose that x y K g ϕ ( K ) . Then there is k K g with x y k ϕ ( K ) , and then x y ( m + k ) ϕ ( K ) . Since x y ( m + k ) = x y m + x y k K and x y ( K : R M ) , we conclude that x ( m + k ) K or y ( m + k ) K . So, x m K or y m K , which is a contradiction. Thus x y K ϕ ( K ) .
  • Suppose that x ( K : R e M ) m ϕ ( K ) . Then there is a ( K : R e M ) with x a m ϕ ( K ) , and then x ( y + a ) m ϕ ( K ) as x y m ϕ ( K ) . Since a m K , we obtain that x ( y + a ) m K . Then x m K or ( y + a ) m K or x ( y + a ) ( K : R M ) . Hence x m K or y m K or x y ( K : R M ) , which is a contradiction. Hence, x ( K : R e M ) m ϕ ( K ) .
  • It can be easily proved by using a similar procedure in part (2).
  • Assume that ( K : R e M ) 2 m ϕ ( K ) . Then there exist a , b ( K : R e M ) with a b m ϕ ( K ) , and then by parts (2) and (3), ( x + a ) ( y + b ) m ϕ ( K ) . Clearly, ( x + a ) ( y + b ) m K . Then ( x + a ) m K or ( y + b ) m K or ( x + a ) ( y + b ) ( K : R M ) . Therefore, x m K or y m K or x y ( K : R M ) , which is a contradiction. Consequently, ( K : R e M ) 2 m ϕ ( K ) .
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 R e and m M g with x y m ϕ ( K ) , x y ( K : R M ) , x m K and y m 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 G and K be a g-ϕ-2-abs. prim. submodule of M. Consider x , y R e and m M g with x y m ϕ ( K ) , x y ( K : R M ) , x m G r a d M ( K ) and y m G r a d M ( K ) . Then
1. 
x y K g ϕ ( K ) .
2. 
x ( K : R e M ) m ϕ ( K ) .
3. 
y ( K : R e M ) m ϕ ( K ) .
4. 
( K : R e M ) 2 m ϕ ( K ) .
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 R e and m M g with x y m ϕ ( K ) , x y ( K : R M ) , x m G r a d M ( K ) and y m G r a d M ( K ) . 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 G . If K is a g-ϕ-2-abs. prim. submodule of M that is not g-2-abs. prim., then ( K : R e M ) 2 K g ϕ ( K ) .
Proof. 
Since K is a g- ϕ -2-abs. prim. submodule of M that is not g-2-abs. prim., there are x , y R e and m M g such that x y m ϕ ( K ) , x y ( K : R M ) , x m G r a d M ( K ) and y m G r a d M ( K ) . Suppose that ( K : R e M ) 2 K g ϕ ( K ) . Then there are a , b ( K : R e M ) and k K g such that a b k ϕ ( K ) . By Theorem 10, we get ( x + a ) ( y + b ) ( m + k ) K ϕ ( K ) . So, ( x + a ) ( m + k ) G r a d M ( K ) or ( y + b ) ( m + k ) G r a d M ( K ) or ( x + a ) ( y + b ) ( K : R M ) . Therefore, x m G r a d M ( K ) or y m G r a d M ( K ) or x y ( K : R M ) , which is a contradiction. Hence, ( K : R e M ) 2 K g ϕ ( K ) . □

4. Conclusions

In this study, we introduced the notions of graded ϕ -2-abs. and graded ϕ -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 ϕ -r-ideals as a generalization of the concept of graded r-ideals.

Author Contributions

Conceptualization, A.S.A., M.B. and R.A.-D.; methodology, A.S.A., M.B. and R.A.-D.; validation, A.S.A., M.B. and R.A.-D.; investigation, A.S.A., M.B. and R.A.-D.; writing—original draft preparation, A.S.A., M.B. and R.A.-D.; software, A.S.A., M.B. and R.A.-D.; resources, A.S.A., M.B. and R.A.-D.; writing—review and editing, A.S.A., M.B. and R.A.-D. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors gratefully thank the referees for the constructive comments, corrections and suggestions which definitely help to improve the readability and quality of the article.

Conflicts of Interest

The authors declare that they have no conflict of interest.

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Alshehry, A.S.; Bataineh, M.; Abu-Dawwas, R. Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules. Mathematics 2021, 9, 1083. https://doi.org/10.3390/math9101083

AMA Style

Alshehry AS, Bataineh M, Abu-Dawwas R. Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules. Mathematics. 2021; 9(10):1083. https://doi.org/10.3390/math9101083

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Alshehry, Azzh Saad, Malik Bataineh, and Rashid Abu-Dawwas. 2021. "Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules" Mathematics 9, no. 10: 1083. https://doi.org/10.3390/math9101083

APA Style

Alshehry, A. S., Bataineh, M., & Abu-Dawwas, R. (2021). Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules. Mathematics, 9(10), 1083. https://doi.org/10.3390/math9101083

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