aa r X i v : . [ m a t h . A C ] F e b On Graded φ − Prime 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. 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 isa homogeneous element of R and m is a homogeneous element of M such that rm ∈ K − φ ( K ), then either m ∈ K or r ∈ ( K : R M ). If φ ( K ) = ∅ for all K ∈ GS ( M ), then a graded φ − prime submodule is exactly a graded prime sub-module. If φ ( K ) = { } for all K ∈ GS ( M ), then a graded φ − prime submodule isexactly a graded weakly prime submodule. Several properties of graded φ − primesubmodules have been investigated.
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 .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 [
5, 6 ]. Mathematics Subject Classification.
Primary 16W50; Secondary 13A02.
Key words and phrases.
Graded prime submodules; graded φ -prime submodules. ALSHEHRY
AND RASHID
ABU-DAWWAS
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 . Let I be a proper graded ideal of R . Then the graded radical of I is Grad ( I ), andis defined to be the set of all r ∈ R such that for each g ∈ G , there exists a positiveinteger n g for which r n g g ∈ I . One can see that if r ∈ h ( R ), then r ∈ Grad ( I ) if andonly if r n ∈ I for some positive integer n . In fact, Grad ( I ) is a graded ideal of R ,see [ ]. A graded R -submodule K of M is called a graded radical R -submodule of M if Grad (( K : R M )) = ( K : R M ).Graded prime ideals play a fundamental role in graded ring theory. One of thenatural generalizations of graded prime ideals which have attracted the interest ofseveral authors in the last two decades is the concept of graded prime submodules,see for example [
1, 3, 4, 7 ]. A proper graded R -submodule K of M is said to bea graded prime R -submodule of M if whenever r ∈ h ( R ) and m ∈ h ( M ) such that rm ∈ K , then either m ∈ K or r ∈ ( K : R M ). It is known that if K is a gradedprime R -submodule of M , then ( K : R M ) is a graded prime ideal of R . A propergraded 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 either m ∈ K or r ∈ ( K : R M ). A proper graded R -submodule K of M is said to be a gradedalmost prime R -submodule of M if whenever r ∈ h ( R ) and m ∈ h ( M ) such that rm ∈ K − ( K : R M ) K , then either m ∈ K or r ∈ ( K : R M ). So, every gradedprime submodule is a graded weakly prime submodule, and every graded weaklyprime submodule is a graded almost prime submodule. Let GS ( M ) be the set ofall graded R -submodules of M and φ : GS ( M ) → GS ( M ) S {∅} be a function. Aproper graded R -submodule K of M is said to be a graded φ − prime R -submoduleof M if whenever r ∈ h ( R ) and m ∈ h ( M ) such that rm ∈ K − φ ( K ), then either m ∈ K or r ∈ ( K : R M ). Since K − φ ( K ) = K − ( K T φ ( P )), so without loss ofgenerality, throughout this article, we will consider φ ( K ) ⊆ K . Let M be a G -graded R -module, g ∈ G and K be a graded R -submodule of M such that K g = M g . Then K is said to be a g − φ − prime R -submodule of M if whenever r ∈ R e and m ∈ M g such that rm ∈ K − φ ( K ), then either m ∈ K or r ∈ ( K : R M ). Throughout thisarticle, we use the following functions: φ ∅ ( K ) = ∅ for all K ∈ GS ( M ), φ ( K ) = { } for all K ∈ GS ( M ), φ ( K ) = ( K : R M ) K for all K ∈ GS ( M ), φ ( K ) = ( K : R M ) K for all K ∈ GS ( M ) and φ w ( K ) = ∞ \ i =1 ( K : R M ) i K for all K ∈ GS ( M ). N GRADED φ − PRIME SUBMODULES 3
Clearly, graded φ ∅ − prime, φ − prime and φ − prime submodules are gradedprime, graded weakly prime and graded almost prime submodules respectively. Ob-viously, for any graded submodule and every positive integer k , we have the followingimplications: graded prime submodule ⇒ graded φ w − prime submodule ⇒ graded φ k − prime submodule ⇒ graded φ k − − prime submodule. For functions φ, ϕ : GS ( M ) → GS ( M ) S {∅} , we write φ ≤ ϕ if φ ( K ) ⊆ ϕ ( K )for all K ∈ GS ( M ). So, if φ ≤ ϕ , then every graded φ − prime submodule is graded ϕ − prime.Among several results, we proved that if K is a g − φ − prime R -submodule of M such that ( K : R e M ) K g * φ ( K ), then K is a g -prime R -submodule of M (Theorem2.3). We showed that if K is a g − φ − prime R -submodule of M which is not g -primeand φ ( K ) ⊆ ( K : R e M ) K g , then φ ( K ) = ( K : R e M ) i K g for all i ≥ = m ∈ M g such that R e m = M g , (0 : R m ) = { } and Rm is not a g -prime R -submodule of M , then Rm is not a g − φ − prime R -submodule of M (Theorem 2.6). We showed that if r ∈ R e such that rM g = M g and (0 : M r ) ⊆ rM , then rM is a g − φ − prime R -submodule of M if and only ifit is a g -prime R -submodule of M (Theorem 2.8). In Theorem 2.9, we introduceda characterization for graded g − φ − prime R -submodules. In Theorem 2.10, westudy graded φ − prime R -submodules over graded quotient R -modules. In Theorem2.11, we examine graded φ − prime R -submodules over multiplicative subsets of h ( R ).Finally, we proposed an interesting question (Question 2.12). Most of the results inthis article are inspired from [ ].
2. Graded φ − Prime Submodules
In this section, we introduce and study the concept of graded φ − prime submod-ules. 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 − prime R -submodule of M if whenever r ∈ h ( R ) and m ∈ h ( M ) such that rm ∈ K − φ ( K ) , then either m ∈ K or r ∈ ( 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 − prime R -submodule of M if whenever r ∈ R e and m ∈ M g such that rm ∈ K − φ ( K ) , then either m ∈ K or r ∈ ( K : R M ) . The concept of g -prime ideals have been introduced and investigated in [ ]. Let P be a graded ideal of a G -graded ring R and g ∈ G such that P g = R g . Then P is called a g -prime ideal of R if whenever x, y ∈ R g such that xy ∈ P , then either x ∈ P or y ∈ P . Motivated by g -prime ideals, we introduce the following definitions: Definition . Let M be a G -graded R -module, K be a graded R -submoduleof M and g ∈ G such that K g = M g . Then (1) K is said to a g -prime R -submodule of M if whenever r ∈ R e and m ∈ M g such that rm ∈ K , then either m ∈ K or r ∈ ( K : R M ) . (2) K is said to a g -weakly prime R -submodule of M if whenever r ∈ R e and m ∈ M g such that = rm ∈ K , then either m ∈ K or r ∈ ( K : R M ) . AZZH SAAD
ALSHEHRY
AND RASHID
ABU-DAWWAS
Theorem . Let M be a G -graded R -module, g ∈ G and K be a g − φ − prime R -submodule of M . If ( K : R e M ) K g * φ ( K ) , then K is a g -prime R -submodule of M . Proof.
Let r ∈ R e and m ∈ M g such that rm ∈ K . If rm / ∈ φ ( K ), then since K is g − φ − prime, we have r ∈ ( K : R M ) or m ∈ K . Suppose that rm ∈ φ ( K ). If rK g * φ ( K ), then there exists p ∈ K g such that rp / ∈ φ ( K ), and then r ( m + p ) ∈ K − φ ( K ). Therefore, r ∈ ( K : R M ) or m + p ∈ K , and hence r ∈ ( K : R M )or m ∈ K . So, we may assume that rK g ⊆ φ ( K ). If ( K : R e M ) m * φ ( K ), thenthere exists u ∈ ( K : R e M ) such that um / ∈ φ ( K ), and then ( r + u ) m ∈ K − φ ( K ).Since K is a g − φ − prime submodule, we have r + u ∈ ( K : R M ) or m ∈ K .So, r ∈ ( K : R M ) or m ∈ K . Therefore, we assume that ( K : R e M ) m ⊆ φ ( K ).Now, since ( K : R e M ) K g * φ ( K ), there exist s ∈ ( K : R e M ) and t ∈ K g such that st / ∈ φ ( K ). So, ( r + s )( m + t ) ∈ K − φ ( K ), and hence r + s ∈ ( K : R M ) or m + t ∈ K .Therefore, r ∈ ( K : R M ) or m ∈ K . (cid:3) Corollary . Let M be a G -graded R -module, g ∈ G and K be a g -weaklyprime R -submodule of M . If ( K : R e M ) K g = { } , then K is a g -prime R -submoduleof M . Proof.
Apply Theorem 2.3 by φ = φ . (cid:3) Corollary . Let M be a G -graded R -module, g ∈ G and K be a g − φ − prime R -submodule of M which is not g -prime. If φ ( K ) ⊆ ( K : R e M ) K g , then φ ( K ) =( K : R e M ) i K g for all i ≥ . Proof.
Since K is not a g -prime R -submodule of M , we have by Theorem2.3, ( K : R e M ) K g ⊆ φ ( K ) ⊆ ( K : R e M ) K g ⊆ ( K : R e M ) K g , which implies that φ ( K ) = ( K : R e M ) K g = ( K : R e M ) K g . Hence, φ ( K ) = ( K : R e M ) i K g for all i ≥ (cid:3) Theorem . Let M be a G -graded R -module, g ∈ G and = m ∈ M g suchthat R e m = M g and (0 : R m ) = { } . If Rm is not a g -prime R -submodule of M ,then Rm is not a g − φ − prime R -submodule of M . Proof.
Since Rm is not a g -prime R -submodule of M , there exist a ∈ R e and y ∈ M g such that a / ∈ ( Rm : R M ), y / ∈ Rm and ay ∈ Rm . If ay / ∈ ( Rm : R M ) m ,then Rm is not a g − φ − prime R -submodule of M . Let ay ∈ ( Rm : R M ) m .Then y + m / ∈ Rm and a ( y + m ) ∈ Rm . If a ( y + m ) / ∈ ( Rm : R M ) m , then Rm is not a g − φ − prime R -submodule of M . Let a ( y + m ) ∈ ( Rm : R M ) m , then am ∈ ( Rm : R M ) m , which gives that am = rm for some r ∈ ( Rm : R M ). Since(0 : R m ) = { } , it gives that a = r ∈ ( Rm : R M ), which is a contradiction. (cid:3) Corollary . Let M be a G -graded R -module, g ∈ G and m be a non-zeroelement of M g such that (0 : R m ) = { } and R e m = M g . Then Rm is a g -prime R -submodule of M if and only if Rm is a g − φ − prime R -submodule of M . Theorem . Let M be a G -graded R -module, r ∈ R e and g ∈ G such that rM g = M g . If (0 : M r ) ⊆ rM , then rM is a g − φ − prime R -submodule of M ifand only if it is a g -prime R -submodule of M . Proof.
Suppose that rM is a g − φ − prime R -submodule of M . Let b ∈ R e and m ∈ M g such that bm ∈ rM . If bm / ∈ ( rM : R M ) rM , then b ∈ ( rM : R M ) or m ∈ rM , since rM is a g − φ − prime submodule. Suppose that bm ∈ ( rM : R M ) rM . N GRADED φ − PRIME SUBMODULES 5
Now, ( b + r ) m ∈ rM . If ( b + r ) m / ∈ ( rM : R M ) rM , then since rM is a g − φ − primesubmodule, b + r ∈ ( rM : R M ) or m ∈ rM , which gives that b ∈ ( rM : R M ) or m ∈ rM . Assume that ( b + r ) m ∈ ( rM : R M ) rM . Then bm ∈ ( rM : R M ) rM givesthat rm ∈ ( rM : R M ) rM . Hence, there exists y ∈ ( rM : R M ) M such that rm = ry and so m − y ∈ (0 : M r ). This gives that m ∈ ( rM : R M ) M + (0 : M r ) ⊆ rM + (0 : M r ) ⊆ rM . The converse is clear. (cid:3) Theorem . Let M be a G -graded R -module, g ∈ G and K be a graded R -submodule of M such that K g = M g . Then the following statements are equivalent: (1) K is a g − φ − prime R -submodule of M . (2) For m ∈ M g − K , ( K : R e m ) = ( K : R e M ) S ( φ ( K ) : R e m ) . (3) For m ∈ M g − K , ( K : R e m ) = ( K : R e M ) or ( K : R e m ) = ( φ ( K ) : R e m ) . (4) For any ideal I of R e and any graded R -submodule N of M , if IN g ⊆ K and IN g * φ ( K ) , then I ⊆ ( K : R M ) or N g ⊆ K . Proof. (1) ⇒ (2) : Let m ∈ M g − K and r ∈ ( K : R e m ) − ( φ ( K ) : R e m ).Then rm ∈ K − φ ( K ). Since K is a g − φ − prime R -submodule of M , r ∈ ( K : R e M ). As we may assume that φ ( K ) ⊆ K , the other inclusionalways holds.(2) ⇒ (3) : It is known that if a subgroup is the union of two subgroups,then it is equal to one of them.(3) ⇒ (4) : Let I be an ideal of R e and N be a graded R -submoduleof M such that IN g ⊆ K . Suppose that I * ( K : R M ) and N g * K .We show that IN g ⊆ φ ( K ). Let r ∈ I and m ∈ N g . If r / ∈ ( K : R e M ),then since rm ∈ K , we have ( K : R e m ) = ( K : R e M ). Hence, by ourassumption, ( K : R e m ) = ( φ ( K ) : R e m ). So, rm ∈ φ ( K ). Now, assume that r ∈ I T ( K : R e M ). Let u ∈ I − ( K : R e M ). Then r + u ∈ I − ( K : R e M ). So,by the first case, for each m ∈ N g we have um ∈ φ ( K ) and ( r + u ) m ∈ φ ( K ).This gives that rm ∈ φ ( K ). Thus in any case rm ∈ φ ( K ). Therefore, IN g ⊆ φ ( K ).(4) ⇒ (1) : Let r ∈ R e and m ∈ M g such that rm ∈ K − φ ( K ).Suppose that I = R e r and N = Rm . Then I is an ideal of R e and N isa graded R -submodule of M such that IN g = R e r ( Rm ) g = R e rR e m g = R e rR e m = R e R e rm ⊆ R e K ⊆ RK ⊆ K and IN g * φ ( K ). By ourassumption, I ⊆ ( K : R M ) or N g ⊆ K , and then r ∈ ( K : R M ) or m ∈ N T M g = N g ⊆ K . (cid:3) Let M be a G -graded R -module and K be a graded R -submodule of M . Then M/K is a G -graded R -module by ( M/K ) g = ( M g + K ) /K for all g ∈ G . Let φ : GS ( M ) → GS ( M ) S {∅} be a function. Define φ K : GS ( M/K ) → GS ( M/K ) S {∅} by φ K ( N/K ) = ( φ ( N ) + K ) /K for N ⊇ K and φ K ( N/K ) = ∅ for φ ( N ) = ∅ . Theorem . Let M be a graded R -module and K be a graded R -submoduleof M . If N is a graded φ − prime R -submodule of M and K ⊆ N , then N/K is agraded φ K − prime R -submodule of M/K . Proof.
By ([ ], Lemma 3.2), N/K is a graded R -submodule of M/K . Let r ∈ h ( R ) and m + K ∈ h ( M/K ) such that r ( m + K ) = rm + K ∈ N/K − φ K ( N/K ).Then m ∈ h ( M ) such that rm ∈ N − ( φ ( N ) + K ), and then rm ∈ N − φ ( N ). Since N is graded φ − prime, r ∈ ( N : R M ) or m ∈ N , and then r ∈ ( N/K : R M/K ) or m ∈ N . Therefore, N/K is a graded φ K − prime R -submodule of M/K . (cid:3) AZZH SAAD
ALSHEHRY
AND RASHID
ABU-DAWWAS
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 is agraded R -submodule of M , then S − K is a graded S − R -submodule of S − M . It iswell known that there is a one-to-one correspondence between the set of all gradedprime R -submodules K of M with ( K : R M ) T S = ∅ and the set of all gradedprime S − R -submodules S − K of S − M . For a graded R -submodule K of M , K ( S ) = { m ∈ M : there exists s ∈ S such that sm ∈ K } is a graded R -submoduleof M containing K and S − ( K ( S )) = S − K . Let φ : GS ( M ) → GS ( M ) S {∅} bea function. Define ( S − φ ) : GS ( S − M ) → GS ( S − M ) S {∅} by ( S − φ )( S − K ) = S − ( φ ( K ( S )) if φ ( K ( S )) = ∅ and ( S − φ )( S − K ) = ∅ if φ ( K ( S )) = ∅ . Note that,( S − φ ∅ ) = φ ∅ and ( S − φ ) = φ . Theorem . Let M be a G -graded R -module and K be a graded φ − prime R -submodule of M . Suppose that S is a multiplicative subset of h ( R ) . If S − K = S − M and S − ( φ ( K )) ⊆ ( S − φ )( S − K ) , then S − K is a graded ( S − φ ) − prime S − R -submodule of S − M . Moreover, if g ∈ G such that ( S − K ) T M g = S − (( φ ( K )) ,then K ( S ) T M g = K . Proof.
Let r/s ∈ h ( S − R ) and m/t ∈ h ( S − M ) such that ( r/s )( m/t ) ∈ S − K − ( S − φ )( S − K ). Then rm/st ∈ S − K − S − ( φ ( K )), and then there ex-ists u ∈ S such that urm ∈ K − φ ( K ) (note that, for each v ∈ S , vrm / ∈ φ ( K )).Since K is graded φ − prime and ( K : R M ) T S = ∅ , it gives that rm ∈ K − φ ( K ) andso r ∈ ( K : R M ) or m ∈ K , and then r/s ∈ S − ( K : R M ) ⊆ ( S − K : S − R S − M ) or m/t ∈ S − K . Hence, S − K is a graded ( S − φ ) − prime S − R -submodule of S − M .Moreover, suppose that ( S − K ) T M g = S − (( φ ( K )) for some g ∈ G and m ∈ K ( S ) T M g . Then there exists s ∈ S such that sm ∈ K . If sm / ∈ φ ( K ), then m ∈ K .If sm ∈ φ ( K ), then m ∈ φ ( K )( S ). So, K ( S ) T M g = K S ( φ ( K )( S )). Hence, either K ( S ) T M g = K or K ( S ) T M g = ( φ ( K )( S )). If K ( S ) T M g = ( φ ( K )( S )), then wehave ( S − K ) T M g = ( S − K ( S )) T M g = S − ( φ ( K )( S )) = S − ( φ ( K )), which is notthe case. So, K ( S ) T M g = K . (cid:3) Question . Let M be a graded R -module, S be a multiplicative subset of h ( R ) and K be a graded R -submodule of M . If S − K is a graded S − φ − prime S − R -submodule of S − M , then clearly, ( K : R M ) T S = ∅ . In general, under whatconditions K will be a graded φ − prime R -submodule of M ?. Even in the cases φ = φ , φ = φ and φ = φ , we could not answer this question. 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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