On Generalizations of Graded 2-absorbing and Graded 2-absorbing primary submodules
aa r X i v : . [ m a t h . R A ] F e b On Generalizations of Graded -absorbing andGraded -absorbing primary submodules Azzh Saad
Alshehry and Rashid
Abu-Dawwas
Abstract.
Let R be a graded commutative ring with non-zero unity 1 and M bea 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 ofthe 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 ) → GS ( M ) S {∅} be a function. A proper graded R -submodule K of M is said tobe a graded φ -2-absorbing R -submodule of M if whenever x, y are homogeneouselements of R and m is a homogeneous element of M with xym ∈ K − φ ( K ), then xm ∈ K or ym ∈ K or xy ∈ ( K : R M ), and K is said to be a graded φ -2-absorbingprimary R -submodule of M if whenever x, y are homogeneous elements of R and m is a homogeneous element of M with xym ∈ K − φ ( K ), then xm or ym is inthe graded radical of K or xy ∈ ( K : R M ). We investigate several properties ofthese new types of graded submodules.
1. Introduction
Throughout this article, G will be a group with identity e and R a commutativering with nonzero unity 1. Then R is said to be G -graded if R = M g ∈ G R g with R g R h ⊆ R gh 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 bewritten uniquely as X 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 tobe G -graded if M = M g ∈ G M g with R g M h ⊆ M gh for all g, h ∈ G where M g is anadditive subgroup of M for all g ∈ G . The elements of M g are called homogeneousof 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 X g ∈ G x g , where x g is the component of x in M g . Mathematics Subject Classification.
Primary 16W50; Secondary 13A02.
Key words and phrases.
Graded φ -prime submodule, graded φ -primary submodule, graded2-absorbing primary submodule, graded weakly 2-absorbing primary submodule, graded φ -2-absorbing submodule, graded φ -2-absorbing primary submodule. ALSHEHRY
AND RASHID
ABU-DAWWAS
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 -submoduleif K = M g ∈ G ( K ∩ M g ), i.e., for x ∈ K , x = 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 andterminology, see [
12, 15 ]. Lemma . ( [ ] , Lemma 2.1) Let R be a graded ring and M be a graded R -module. (1) If I and J are graded ideals of R , then I + J and I T J are graded ideals of R . (2) If N and K are graded R -submodules of M , then N + K and N T K aregraded R -submodules of M . (3) If N is a graded R -submodule of M , r ∈ h ( R ) , x ∈ h ( M ) and I is a gradedideal of R , then Rx , IN and rN are graded R -submodules of M . Moreover, ( N : R M ) = { r ∈ R : rM ⊆ N } is a graded ideal of R . In particular,
Ann R ( M ) = (0 : R M ) is a graded ideal of R . Let I be a propergraded ideal of R . Then the graded radical of I is Grad ( I ), and is defined to bethe set of all r ∈ R such that for each g ∈ G , there exists a positive integer n g forwhich r n g g ∈ I . One can see that if r ∈ h ( R ), then r ∈ Grad ( I ) if and only if r n ∈ I for some positive integer n . In fact, Grad ( I ) is a graded ideal of R , see [ ]. Let N be a graded R -submodule of M . Then the graded radical of N is denoted by Grad M ( N ) and it is defined to be the intersection of all graded prime submodulesof M containing N . If there is no graded prime submodule containing N , then wetake Grad M ( N ) = M .A graded prime (resp. graded primary) R -submodule is a proper graded R -submodule K of M with the property that for r ∈ h ( R ) and m ∈ h ( M ) such that rm ∈ K implies that m ∈ K or r ∈ ( K : R M ) (resp. m ∈ K or r ∈ Grad (( 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([
1, 3, 5, 9, 16 ]). Graded weakly prime submodules have been introduced by Ataniin [ ]. A proper graded R -submodule K of M is said to be a graded weakly prime R -submodule of M if whenever r ∈ h ( R ) and m ∈ h ( M ) such that 0 = rm ∈ K ,then m ∈ K or r ∈ ( K : R M ). The concept of graded φ -prime submodules hasbeen introduced in [ ]. Let GS ( M ) be the set of all graded R -submodules of M and φ : GS ( M ) → GS ( M ) S {∅} be a function. A proper graded R -submodule K of M is said to be a graded φ -prime R -submodule of M if whenever r ∈ h ( R ) and m ∈ h ( M ) such that rm ∈ K − φ ( K ), then m ∈ K or r ∈ ( K : R M ).The concept of graded 2-absorbing ideals (resp. graded weakly 2-absorbingideals) is introduced in [ ] as a different generalization of graded prime ideals (resp.graded weakly prime ideals). A proper graded ideal I of R is a graded 2-absorbingideal (resp. graded weakly 2-absorbing ideal) of R if whenever x, y, z ∈ h ( R ) and xyz ∈ I (resp. 0 = xyz ∈ I ), then xy ∈ I or xz ∈ I or yz ∈ I . Then introduc-ing graded 2-absorbing submodules (resp. graded weakly 2-absorbing submodules)in [ ] generalized the concept of graded 2-absorbing ideals (resp. graded weakly 2-absorbing ideals) to graded submodules as following: A proper graded R -submodule N GENERALIZATIONS OF GRADED 2-ABSORBING AND GRADED 2-ABSORBING PRIMARY SUBMODULES3 K of M is said to be a graded 2-absorbing R -submodule (resp. graded weakly 2-absorbing R -submodule) of M if whenever x, y ∈ h ( R ) and m ∈ h ( M ) with xym ∈ K (resp. 0 = xym ∈ K ), then xy ∈ ( K : R M ) or xm ∈ K or ym ∈ K .Al-Zoubi and Sharafat in [ ] introduced the concept of graded 2-absorbing pri-mary ideals, where a proper graded ideal I of R is called graded 2-absorbing pri-mary if whenever x, y, z ∈ h ( R ) with xyz ∈ I , then xy ∈ I or xz ∈ Grad ( I ) or yz ∈ Grad ( I ). The concept of graded 2-absorbing primary submodules is studiedin [ ] 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 xy ∈ h ( R )and m ∈ h ( M ) with xym ∈ K (resp. 0 = xym ∈ K ), then xy ∈ ( K : R M ) or xm ∈ Grad M ( K ) or ym ∈ Grad M ( K ).A graded R -module M is said to be graded multiplication if for every graded R -submodule N of M , N = IM for some graded deal I of R . In this case, it isknown that I = ( N : R M ). Graded multiplication modules were firstly introducedand studied by Escoriza and Torrecillas in [ ], and further results were obtainedby several authors, see for example [ ]. Let N and K be graded R -submodulesof a graded multiplication R -module M with N = IM and K = J M for somegraded ideals I and J of R . The product of N and K is denoted by N K is definedby
N K = IJ M . Then the product of N and K is independent of presentationsof N and K . In fact, as IJ is a graded ideal of R (see [ ]), N K is a graded R -submodule of M and N K ⊆ N T K . Moreover, for x, y ∈ h ( M ), by xy , wemean the product of Rx and Ry . Also, it is shown in ([ ], Theorem 9) that if N is a proper graded R -submodule of a graded multiplication R -module M , then Grad M ( N ) = Grad (( N : R M )) M .In this article, our aim is to extend the concept of graded 2-absorbing submodulesto graded φ -2-absorbing submodules in completely different way from [ ], and alsoto extend graded 2-absorbing primary submodules to graded φ -2-absorbing primarysubmodules. Our study is inspired from [ ].
2. Graded φ - -Absorbing and Graded φ - -Absorbing PrimarySubmodules In this section, we introduce and study the concepts of graded φ -2-absorbing andgraded φ -2-absorbing primary submodules. Definition . Let M be a G -graded R -module and φ : GS ( M ) → GS ( M ) S {∅} be a function. (1) A proper graded R -submodule K of M is said to be a graded φ -primary R -submodule of M if whenever r ∈ h ( R ) and m ∈ h ( M ) with rm ∈ K − φ ( K ) ,then m ∈ K or r ∈ Grad (( K : R M )) . (2) Let K be a graded R -submodule of M and g ∈ G such that K g = M g . Then K is said to be a g - φ -primary R -submodule of M if whenever r ∈ R e and m ∈ M g with rm ∈ K − φ ( K ) , then m ∈ K or r ∈ Grad (( K : R M )) . (3) A proper graded R -submodule K of M is said to be a graded φ - -absorbing R -submodule of M if whenever x, y ∈ h ( R ) and m ∈ h ( M ) with xym ∈ K − φ ( K ) , then xm ∈ K or ym ∈ K or xy ∈ ( K : R M ) . (4) Let K be a graded R -submodule of M and g ∈ G such that K g = M g .Then K is said to be a g - φ - -absorbing R -submodule of M if whenever AZZH SAAD
ALSHEHRY
AND RASHID
ABU-DAWWAS x, y ∈ R e and m ∈ M g with xym ∈ K − φ ( K ) , then xm ∈ K or ym ∈ K or xy ∈ ( K : R M ) . (5) A proper graded R -submodule K of M is said to be a graded φ - -absorbingprimary R -submodule of M if whenever x, y ∈ h ( R ) and m ∈ h ( M ) with xym ∈ K − φ ( K ) , then xm ∈ Grad M ( K ) or ym ∈ Grad M ( K ) or xy ∈ ( K : R M ) . (6) Let K be a graded R -submodule of M and g ∈ G such that K g = M g . Then K is said to be a g - φ - -absorbing primary R -submodule of M if whenever x, y ∈ R e and m ∈ M g with xym ∈ K − φ ( K ) , then xm ∈ Grad M ( K ) or ym ∈ Grad M ( K ) or xy ∈ ( K : R M ) . Remark . (1) Let K be a graded φ -primary R -submodule of a gradedmultiplication R -module M . Then φ ∅ ( K ) = ∅ graded primary submodule, φ ( K ) = { } graded weakly primary submodule, φ ( K ) = K 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 - -absorbing (resp. graded φ - -absorbing primary) R -submodule of a graded multiplication R -module M . Then φ ∅ ( K ) = ∅ graded -absorbing (resp. graded -absorbing primary)submodule, φ ( K ) = { } graded weakly -absorbing (resp. graded weakly -absorbing primary) submodule, φ ( K ) = K graded almost -absorbing (resp. graded almost -absorbing primary) submodule, φ n ( K ) = K n graded n -almost -absorbing (resp. graded n -almost -absorbing primary) submodule, and φ ω ( K ) = ∞ \ n =1 K n graded ω - -absorbing (resp. graded ω - -absorbingprimary) submodule. (3) For functions φ, ϕ : GS ( M ) → GS ( M ) S {∅} , we write φ ≤ ϕ if φ ( K ) ⊆ ϕ ( K ) for all K ∈ GS ( M ) . Thus clearly we have the following order: φ ∅ ≤ φ ≤ φ ω ≤ ... ≤ φ n +1 ≤ φ n ≤ ... ≤ φ ≤ φ . (4) If φ ≤ ϕ , then every graded φ - -absorbing (resp. graded φ - -absorbingprimary) R -submodule is graded ϕ - -absorbing (resp. graded ϕ - -absorbingprimary). (5) Since K − φ ( K ) = K − ( K T φ ( K )) for any graded R -submodule K of M ,without loss of generality, throughout this article, we assume that φ ( K ) ⊆ K . Theorem . Let M be a graded R -module and K be a proper graded R -submodule of M . Then the followings hold: N GENERALIZATIONS OF GRADED 2-ABSORBING AND GRADED 2-ABSORBING PRIMARY SUBMODULES5 (1) K is a graded φ -prime R -submodule of M ⇒ K is a graded φ - -absorbing R -submodule of M ⇒ K is a graded φ - -absorbing primary R -submoduleof M . (2) If M is a graded multiplication R -module and K is a graded φ -primary R -submodule of M , then K is a graded φ - -absorbing primary R -submoduleof M . (3) For graded multiplication R -module M , K is a graded -absorbing R -submoduleof M ⇒ K is a graded weakly -absorbing R -submodule of M ⇒ K is agraded ω - -absorbing R -submodule of M ⇒ K is a graded ( n + 1) -almost -absorbing R -submodule of M ⇒ K is a graded n -almost -absorbing R -submodule of M for all n ≥ ⇒ K is a graded almost -absorbing R -submodule of M . (4) For graded multiplication R -module M , K is a graded -absorbing pri-mary R -submodule of M ⇒ K is a graded weakly -absorbing primary R -submodule of M ⇒ K is a graded ω - -absorbing primary R -submodule of M ⇒ K is a graded ( n + 1) -almost -absorbing primary R -submodule of M ⇒ K is a graded n -almost -absorbing primary R -submodule of M for all n ≥ ⇒ K is a graded almost -absorbing primary R -submodule of M . (5) Suppose that
Grad M ( K ) = K . Then K is a graded φ - -absorbing primary R -submodule of M if and only if K is a graded φ - -absorbing R -submoduleof M . (6) If M is a graded multiplication R -module and K is an idempotent R -submodule of M , then K is a graded ω - -absorbing R -submodule of M , and K is a graded n -almost -absorbing R -submodule of M for every n ≥ . (7) Let M be a graded multiplication R -module. Then K is a graded n -almost -absorbing (resp. graded n -almost -absorbing primary) R -submodule of M for all n ≥ if and only if K is a graded ω - -absorbing (resp. graded ω - -absorbing primary) R -submodule of M . Proof. (1) It is straightforward.(2) Let x, y ∈ h ( R ) and m ∈ h ( M ) such that xym ∈ K − φ ( K ). Assume that ym / ∈ Grad M ( K ). Then ym / ∈ K and then x ∈ Grad (( K : R M )) as K isa graded φ -primary R -submodule. Therefore, xm ∈ Grad (( K : R M ) M = Grad M ( K ). Consequently, K is graded φ -2-absorbing primary.(3) It is clear by Remark 2.2 (4).(4) It is clear by Remark 2.2 (4).(5) The claim is obvious.(6) 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-absorbing R -submoduleof M . By (3), we conclude that K is a graded n -almost 2-absorbing R -submodule of M for all n ≥ K is a graded n -almost 2-absorbing (resp. graded n -almost2-absorbing primary) R -submodule of M for all n ≥
2. Let x, y ∈ h ( R ) and m ∈ h ( M ) with xym ∈ K but xym / ∈ ∞ \ n =1 K n . Hence xym / ∈ K n for some n ≥
2. Since K is graded n -almost 2-absorbing (resp. graded n -almost2-absorbing primary) for all n ≥
2, this implies either xy ∈ ( K : R M )or ym ∈ K or xm ∈ K (resp. xy ∈ ( K : R M ) or ym ∈ Grad M ( K ) or AZZH SAAD
ALSHEHRY
AND RASHID
ABU-DAWWAS xm ∈ Grad M ( K )). This completes the first implication. The converse isclear from (3) (resp. from (4)). (cid:3) 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 ([ ]). Lemma . ( [ ] , Lemma 3.2) Let M be a graded R -module, K be a graded R -submodule of M , and N be an R -submodules of M such that K ⊆ N . Then N isa graded R -submodule of M if and only if N/K is a graded R -submodule of M/K . Theorem . Let M be a graded R -module and K be a proper graded R -submodule of M . Then the following hold: (1) K is a graded φ - -absorbing R -submodule of M if and only if K/φ ( K ) is agraded weakly -absorbing R -submodule of M/φ ( K ) . (2) K is a graded φ - -absorbing primary R -submodule of M if and only if K/φ ( K ) is a graded weakly -absorbing primary R -submodule of M/φ ( K ) . (3) K is a graded φ -prime R -submodule of M if and only if K/φ ( K ) is a gradedweakly prime R -submodule of M/φ ( K ) . (4) K is a graded φ -primary R -submodule of M if and only if K/φ ( K ) is agraded weakly primary R -submodule of M/φ ( K ) . Proof. (1) If φ ( K ) = ∅ , then it is done. Suppose that φ ( K ) = ∅ . Let x, y ∈ h ( R ) and m + φ ( K ) ∈ h ( M/φ ( K )) such that φ ( K ) = xy ( m + φ ( K )) = xym + φ ( K ) ∈ K/φ ( K ). Then m ∈ h ( M ) such that xym ∈ K , but xym / ∈ φ ( K ). Hence either xy ∈ ( K : R M ) or ym ∈ K or xm ∈ K . So, xy ∈ ( 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 xym ∈ K and xym / ∈ φ ( K ). Then m + φ ( K ) ∈ h ( M/φ ( K ) such that φ ( K ) = xy ( m + φ ( K )) ∈ K/φ ( K ). Hence xy ∈ ( K/φ ( K ) : R M/φ ( K )) or y ( m + φ ( K )) ∈ K/φ ( K ) or x ( m + φ ( K )) ∈ K/φ ( K ). So, xy ∈ ( K : R M )or ym ∈ K or xm ∈ K . Thus K is a graded φ -2-absorbing R -submodule of M .(2) Let x, y ∈ h ( R ) and m + φ ( K ) ∈ h ( M/φ ( K )) such that φ ( K ) = xy ( m + φ ( K )) = xym + φ ( K ) ∈ K/φ ( K ). Then m ∈ h ( M ) such that xym ∈ K , but xym / ∈ φ ( K ). Hence either xy ∈ ( K : R M ) or ym ∈ Grad M ( K ) or xm ∈ Grad M ( K ). So, xy ∈ ( K : R M ) or y ( m + φ ( K )) ∈ Grad M ( K ) /φ ( K ) or x ( m + φ ( K )) ∈ Grad M ( K ) /φ ( K ). The result holds since Grad M ( K ) /φ ( K ) = Grad
M/φ ( K ) ( K/φ ( K )). One can easily prove the converse.Similarly, one can easily prove (3) and (4). (cid:3) Let M and L be two G -graded R -modules. An R -homomorphism f : M → L issaid to be a graded R -homomorphism if f ( M g ) ⊆ L g for all g ∈ G ([ ]). Lemma . ( [ ] , Lemma 2.16) Suppose that f : M → L is a graded R -homomorphism. If N is a graded R -submodule of L , then f − ( N ) is a graded R -submodule of M . Lemma . ( [ ] , 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 . Suppose that f : M → L is a graded R -epimorphism. Let φ : GS ( M ) → GS ( M ) S {∅} and ϕ : GS ( L ) → GS ( L ) S {∅} be functions. Then thefollowing hold: N GENERALIZATIONS OF GRADED 2-ABSORBING AND GRADED 2-ABSORBING PRIMARY SUBMODULES7 (1) If N is a graded ϕ - -absorbing primary R -submodule of L and φ ( f − ( N )) = f − ( ϕ ( N )) , then f − ( N ) is a graded φ - -absorbing primary R -submodule of M . (2) If K is a graded φ - -absorbing primary R -submodule of M containing Ker ( f ) and ϕ ( f ( K )) = f ( φ ( K )) , then f ( K ) is a graded ϕ - -absorbing primary R -submodule of L . (3) If N is a graded ϕ - -absorbing R -submodule of L and φ ( f − ( N )) = f − ( ϕ ( N )) ,then f − ( N ) is a graded φ - -absorbing R -submodule of M . (4) If K is a graded φ - -absorbing R -submodule of M containing Ker ( f ) and ϕ ( f ( K )) = f ( φ ( K )) , then f ( K ) is a graded ϕ - -absorbing R -submodule of L . Proof. (1) Since f is epimorphism, f − ( N ) is a proper graded R -submoduleof M . Let x, y ∈ h ( R ) and m ∈ h ( M ) such that xym ∈ f − ( N ) and xym / ∈ f − ( ϕ ( N )). Since xym ∈ f − ( N ), xyf ( m ) ∈ N . Also, φ ( f − ( N )) = f − ( ϕ ( N )) implies that xyf ( m ) / ∈ ϕ ( N ). Thus xyf ( m ) ∈ N − ϕ ( N ). Then xy ∈ ( N : R L ) or xf ( m ) ∈ Grad L ( N ) or yf ( m ) ∈ Grad L ( N ). Thus xy ∈ ( f − ( N ) : R M ) or xm ∈ f − ( Grad L ( N )) or ym ∈ f − ( Grad L ( N )).Since f − ( Grad L ( N )) ⊆ Grad M ( f − ( N )), we conclude that f − ( N ) is agraded φ -2-absorbing primary R -submodule of M .(2) Let x, y ∈ h ( R ) and s ∈ h ( L ) such that xys ∈ f ( K ) − ϕ ( f ( K )). Since f is graded epimorphism, there exists m ∈ h ( M ) such that s = f ( m ).Therefore, f ( xym ) ∈ f ( K ) and so xym ∈ K as Ker ( f ) ⊆ K . Since ϕ ( f ( K )) = f ( φ ( K )), we have xym / ∈ φ ( K ). Hence xym ∈ K − φ ( K ). Itimplies that xy ∈ ( K : R M ) or xm ∈ Grad M ( K ) or ym ∈ Grad M ( K ).Thus xy ∈ ( f ( K ) : R L ) or xs ∈ f ( Grad M ( K )) or ys ∈ f ( Grad M ( K )).Since Ker ( f ) ⊆ K , f ( Grad M ( K )) = Grad L ( f ( K )), and then we are done.Similarly, one can easily prove (3) and (4). (cid:3) Let M be a G -graded R -module and S ⊆ h ( R ) be a multiplicative set. Then S − M is a G -graded S − R -module with ( S − M ) g = (cid:8) ms , m ∈ M h , s ∈ S ∩ R hg − (cid:9) for all g ∈ G , and ( S − R ) g = (cid:8) as , a ∈ R h , s ∈ S ∩ R hg − (cid:9) for all g ∈ G . If K isa graded R -submodule of M , then S − K is a graded S − R -submodule of S − M .Let φ : GS ( M ) → GS ( M ) S {∅} be a function and define φ S : GS ( S − M ) → GS ( S − M ) S {∅} by φ S ( S − K ) = S − φ ( K ), and φ S ( S − K ) = ∅ when φ ( K ) = ∅ forevery graded R -submodule K of M . Theorem . Let M be a graded R -module and S ⊆ h ( R ) be a multiplicativeset. (1) If K is a graded φ - -absorbing primary R -submodule of M and S − K = S − M , then S − K is a graded φ S - -absorbing primary S − R -submodule of S − M . (2) If K is a graded φ - -absorbing R -submodule of M and S − K = S − M , then S − K is a graded φ S - -absorbing S − R -submodule of S − M . Proof. (1) Let x/s , y/s ∈ h ( S − R ) and m/s ∈ h ( S − M ) such that( x/s )( y/s )( m/s ) ∈ S − K − φ S ( S − K ). Then x, y ∈ h ( R ) and m ∈ h ( M )such that uxym ∈ K − φ ( K ) for some u ∈ S , and then uxm ∈ Grad M ( K ) or uym ∈ Grad M ( K ) or xy ∈ ( K : R M ). So, ( x/s )( m/s ) = ( uxm ) / ( us s ) ∈ S − ( Grad M ( K )) ⊆ Grad S − M ( S − K ) or ( y/s )( m/s ) = ( uym ) / ( us s ) ∈ AZZH SAAD
ALSHEHRY
AND RASHID
ABU-DAWWAS
Grad S − M ( S − K ) or ( x/s )( y/s ) = ( xy ) / ( s s ) ∈ S − ( K : R M ) ⊆ ( S − K : S − R S − M ).Similarly, one can easily prove (2). (cid:3) Let M be a G -graded R -module, M be a G -graded R -module and R = R × R . Then M = M × M is G -graded R -module with M g = ( M ) g × ( M ) g forall g ∈ G , where R g = ( R ) g × ( R ) g for all g ∈ G , ([ ]). Lemma . ( [ ] , Lemma 3.12) Let M be a G -graded R -module, M be a G -graded R -module, R = R × R and M = M × M . Then L = N × K is agraded R -submodule of M if and only if N is a graded R -submodule of M and K is a graded R -submodule of M . Lemma . Let M be a G -graded R -module, M be a G -graded R -module, R = R × R and M = M × M . Suppose that ϕ : GS ( M ) → GS ( M ) S {∅} , ϕ : GS ( M ) → GS ( M ) S {∅} be functions and φ = ϕ × ϕ . Assume that K = K × M for some proper graded R -submodule K of M . If K is a graded φ - -absorbing R -submodule of M , then K is a graded ϕ - -absorbing R -submodule of M . Proof.
Let x , y ∈ h ( R ) and m ∈ h ( M ) such that x y m ∈ K − ϕ ( K ).Then for any m ∈ h ( M ), ( x , , ( y , ∈ h ( R ) and ( m , m ) ∈ h ( M ) such that( x , y , m , m ) ∈ ( K × M ) − ( ϕ ( K ) × ϕ ( M )) = K − φ ( K ). Since K is agraded φ -2-absorbing R -submodule of M , we get either ( x , y , ∈ (( K × M ) : R M × M ) or ( x , m , m ) ∈ ( K × M ) or ( y , m , m ) ∈ ( K × M ). So clearly,we conclude that x y ∈ ( K : R M ) or x m ∈ K or y m ∈ K . Therefore, K isa graded ϕ -2-absorbing R -submodule of M . (cid:3) Theorem . Let M be a G -graded R -module, M be a G -graded R -module, R = R × R and M = M × M . Suppose that ϕ : GS ( M ) → GS ( M ) S {∅} , ϕ : GS ( M ) → GS ( M ) S {∅} be functions and φ = ϕ × ϕ . Assume that K = K × M for some proper graded R -submodule K of M . Then the following hold: (1) If ϕ ( M ) = M , then K is a graded φ - -absorbing R -submodule of M ifand only if K is a graded ϕ - -absorbing R -submodule of M . (2) If ϕ ( M ) = M , then K is a graded φ - -absorbing R -submodule of M ifand only if K is a graded -absorbing R -submodule of M . Proof. (1) Suppose that K is a graded ϕ -2-absorbing R -submodule of M . Let x = ( x , x ) , y = ( y , y ) ∈ h ( R ) and m = ( m , m ) ∈ h ( M ) suchthat xym ∈ K − φ ( K ). Since ϕ ( M ) = M , we get that x , y ∈ h ( R )and m ∈ h ( M ) such that x y m ∈ K − ϕ ( K ), and this implies thateither x y ∈ ( K : R M ) or x m ∈ K or y m ∈ K . Thus either xy ∈ ( K : R M ) or xm ∈ K or ym ∈ K . Hence, K is a graded φ -2-absorbing R -submodule of M . The converse holds from Lemma 2.11.(2) Suppose that K is a graded φ -2-absorbing R -submodule of M . Since ϕ ( M ) = M , there exists m ∈ M − ϕ ( M ) and then there exists g ∈ G such that ( m ) g ∈ M − ϕ ( M ). Assume that K is not a graded2-absorbing R -submodule of M . By Lemma 2.11, K is a graded ϕ -2-absorbing R -submodule of M . Hence, there exist x , y ∈ h ( R ) and m ∈ h ( M ) such that x y m ∈ ϕ ( K ), x y / ∈ ( K : R M ), x m / ∈ K and y m / ∈ K . So, ( x , y , m , ( m ) g ) ∈ ( K × M ) − ( ϕ ( K ) × ϕ ( M )) =( K × M ) − φ ( K × M ) which implies that x y ∈ ( K : R M ) or x m ∈ K N GENERALIZATIONS OF GRADED 2-ABSORBING AND GRADED 2-ABSORBING PRIMARY SUBMODULES9 or y m ∈ K , which is a contradiction. Thus K is a graded 2-absorbing R -submodule of M . Conversely, if K is a graded 2-absorbing R -submoduleof M , then K = K × M is a graded 2-absorbing R -submodule of M by([ ], Theorem 3.3). Hence K is a graded φ -2-absorbing R -submodule of M for any φ . (cid:3) Lemma . Let M be a G -graded R -module, M be a G -graded R -module, R = R × R and M = M × M . Suppose that ϕ : GS ( M ) → GS ( M ) S {∅} , ϕ : GS ( M ) → GS ( M ) S {∅} be functions and φ = ϕ × ϕ . Assume that K = K × M for some proper graded R -submodule K of M . If K is a graded φ - -absorbing primary R -submodule of M , then K is a graded ϕ - -absorbing primary R -submodule of M . Proof.
It can be easily proved by using a similar argument in the proof ofLemma 2.11. (cid:3)
Theorem . Let M be a G -graded R -module, M be a G -graded R -module, R = R × R and M = M × M . Suppose that ϕ : GS ( M ) → GS ( M ) S {∅} , ϕ : GS ( M ) → GS ( M ) S {∅} be functions and φ = ϕ × ϕ . Assume that K = K × M for some proper graded R -submodule K of M . Then the following hold: (1) If ϕ ( M ) = M , then K is a graded φ - -absorbing primary R -submoduleof M if and only if K is a graded ϕ - -absorbing primary R -submodule of M . (2) If ϕ ( M ) = M , then K is a graded φ - -absorbing primary R -submodule of M if and only if K is a graded -absorbing primary R -submodule of M . Proof. (1) It can be easily proved by using a similar argument in the proofof Theorem 2.12 (1).(2) Suppose that K is a graded 2-absorbing primary R -submodule of M .Then K = K × M is a graded 2-absorbing primary R -submodule of M by([ ], Theorem 18). Hence K is a graded φ -2-absorbing R -submodule of M for any φ . The remaining of this proof is similar to Theorem 2.12 (2). (cid:3) Theorem . Let M be a G -graded R -module g ∈ G and K be a g - φ -primary R -submodule of M . Suppose that x ∈ R e and m ∈ M g such that xm ∈ φ ( K ) , x / ∈ Grad (( K : R M )) and m / ∈ K . Then (1) xK g ⊆ φ ( K ) . (2) ( K : R e M ) m ⊆ φ ( K ) . (3) ( K : R e M ) K g ⊆ φ ( K ) . Proof. (1) Suppose that xK g * φ ( K ). Then there exists k ∈ K g suchthat xk / ∈ φ ( K ), and then x ( m + k ) / ∈ φ ( K ). Since x ( m + k ) ∈ K and x / ∈ Grad (( 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, xK g ⊆ φ ( K ).(2) Suppose that ( K : R e M ) m * φ ( K ). Then there exists y ∈ ( K : R e M ) suchthat ym / ∈ φ ( K ), and then ( x + y ) m / ∈ φ ( K ) as xm ∈ φ ( K ). Since ym ∈ K ,we get ( x + y ) m ∈ K . Since m / ∈ K , we have that x + y ∈ Grad (( K : R M )).Hence x ∈ Grad (( K : R M )), which is a contradiction.(3) Suppose that ( K : R e M ) K g * φ ( K ). Then there exist y ∈ ( K : R e M ) and k ∈ K g such that yk / ∈ φ ( K ). By (1) and (2), ( x + y )( m + k ) ∈ K − φ ( K ). ALSHEHRY
AND RASHID
ABU-DAWWAS
So, either x + y ∈ Grad (( K : R M )) or m + k ∈ K . Thus we have either x ∈ Grad (( K : R M )) or m ∈ K , which is a contradiction. (cid:3) Remark . Note that if K is a g - φ -primary R -submodule of M which isnot g -primary, then there exist x ∈ R e and m ∈ M g such that xm ∈ φ ( K ) , x / ∈ Grad (( K : R M )) and m / ∈ K . So, every g - φ -primary R -submodule, which is not g -primary, satisfies Theorem 2.15. Theorem . Let M be a G -graded R -module, g ∈ G and K be a g - φ - -absorbing R -submodule of M . Suppose that x, y ∈ R e and m ∈ M g such that xym ∈ φ ( K ) , xy / ∈ ( K : R M ) , xm / ∈ K and ym / ∈ K . Then (1) xyK 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 ) m ⊆ φ ( K ) . Proof. (1) Suppose that xyK g * φ ( K ). Then there exists k ∈ K g with xyk / ∈ φ ( K ), and then xy ( m + k ) / ∈ φ ( K ). Since xy ( m + k ) = xym + xyk ∈ K and xy / ∈ ( K : R M ), we conclude that x ( m + k ) ∈ K or y ( m + k ) ∈ K . So, xm ∈ K or ym ∈ K , which is a contradiction. Thus xyK ⊆ φ ( K ).(2) Suppose that x ( K : R e M ) m * φ ( K ). Then there exists a ∈ ( K : R e M )such that xam / ∈ φ ( K ), and then x ( y + a ) m / ∈ φ ( K ) as xym ∈ φ ( K ). Since am ∈ K , we obtain that x ( y + a ) m ∈ K . Then xm ∈ K or ( y + a ) m ∈ K or x ( y + a ) ∈ ( K : R M ). Hence xm ∈ K or ym ∈ K or xy ∈ ( K : R M ),which is a contradiction. Hence, x ( K : R e M ) m ⊆ φ ( K ).(3) It can be easily proved by using a similar argument in the proof of part (2).(4) Assume that ( K : R e M ) m * φ ( K ). Then there exist a, b ∈ ( K : R e M ) suchthat abm / ∈ 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, xm ∈ K or ym ∈ K or xy ∈ ( K : R M ), which is a contradiction. Consequently, ( K : R e M ) m ⊆ φ ( K ). (cid:3) Remark . Note that if K is a g - φ - -absorbing R -submodule of M which isnot g - -absorbing, then there exist x, y ∈ R e and m ∈ M g such that xym ∈ φ ( K ) , xy / ∈ ( K : R M ) , xm / ∈ K and ym / ∈ K . So, every g - φ - -absorbing R -submodule,which is not g - -absorbing, satisfies Theorem 2.17. Theorem . Let M be a G -graded R -module, g ∈ G and K be a g - φ - -absorbing primary R -submodule of M . Suppose that x, y ∈ R e and m ∈ M g suchthat xym ∈ φ ( K ) , xy / ∈ ( K : R M ) , xm / ∈ Grad M ( K ) and ym / ∈ Grad M ( K ) . Then (1) xyK 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 ) m ⊆ φ ( K ) . Proof.
It can be easily proved by using a similar argument in the proof ofTheorem 2.17. (cid:3)
Remark . Note that if K is a g - φ - -absorbing primary R -submodule of M which is not g - -absorbing primary, then there exist x, y ∈ R e and m ∈ M g such that xym ∈ φ ( K ) , xy / ∈ ( K : R M ) , xm / ∈ Grad M ( K ) and ym / ∈ Grad M ( K ) . So, every N GENERALIZATIONS OF GRADED 2-ABSORBING AND GRADED 2-ABSORBING PRIMARY SUBMODULES11 g - φ - -absorbing primary R -submodule, which is not g - -absorbing primary, satisfiesTheorem 2.19. Theorem . Let M be a G -graded R -module and g ∈ G . If K is a g - φ - -absorbing primary R -submodule of M that is not g - -absorbing primary, then ( K : R e M ) K g ⊆ φ ( K ) . Proof.
Since K is a g - φ -2-absorbing primary R -submodule of M that is not g -2-absorbing primary, there exist x, y ∈ R e and m ∈ M g such that xym ∈ φ ( K ), xy / ∈ ( K : R M ), xm / ∈ Grad M ( K ) and ym / ∈ Grad M ( K ). Suppose that ( K : R e M ) K g * φ ( K ). Then there are a, b ∈ ( K : R e M ) and k ∈ K g such that abk / ∈ φ ( K ).By Theorem 2.19, we get ( x + a )( y + b )( m + k ) ∈ K − φ ( K ). So, ( x + a )( m + k ) ∈ Grad M ( K ) or ( y + b )( m + k ) ∈ Grad M ( K ) or ( x + a )( y + b ) ∈ ( K : R M ). Therefore, xm ∈ Grad M ( K ) or ym ∈ Grad M ( K ) or xy ∈ ( K : R M ), which is a contradiction.Hence, ( K : R e M ) K g ⊆ φ ( K ). (cid:3) Proposition . Let M be a graded R -module and K be a graded φ - -absorbingprimary R -submodule of M . If φ ( K ) is a graded -absorbing primary R -submoduleof M , then K is a graded- -absorbing primary R -submodule of M . Proof.
Assume that x, y ∈ h ( R ) and m ∈ h ( M ) such that xym ∈ K . If xym ∈ φ ( K ), then we conclude that xm ∈ Grad M ( φ ( K )) ⊆ Grad M ( K ) or ym ∈ Grad M ( φ ( K )) ⊆ Grad M ( K ) or xy ∈ ( φ ( K ) : R M ) ⊆ ( K : R M ) since φ ( K ) is graded2-absorbing primary, and so the result holds. If xym / ∈ φ ( K ), then the result holdseasily since K is graded φ -2-absorbing primary. (cid:3) Theorem . Let M be a graded R -module and K be a graded R -submoduleof M with φ ( Grad M ( K )) ⊆ φ ( K ) . If Grad M ( K ) is a graded φ -prime R -submoduleof M , then K is a graded φ - -absorbing primary R -submodule of M . Proof.
Let x, y ∈ h ( R ) and m ∈ h ( M ) such that xym ∈ K − φ ( K ) and xm / ∈ Grad M ( K ). Since Grad M ( K ) is a graded φ -prime R -submodule and xym ∈ Grad M ( K ) − φ ( Grad M ( K )), y ∈ ( Grad M ( K ) : R M ). So, ym ∈ Grad M ( K ). Conse-quently, K is a graded φ -2-absorbing primary R -submodule of M . (cid:3) Acknowledgement
This research was funded by the Deanship of Scientific Research at PrincessNourah bint Abdulrahman University through the Fast-track Research Funding Pro-gram.
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