On Graded classical 2-absorbing second submodules of graded modules over graded commutative rings
aa r X i v : . [ m a t h . A C ] F e b ON GRADED CLASSICAL 2-ABSORBING SECONDSUBMODULES OF GRADED MODULES OVERGRADED COMMUTATIVE RINGS
KHALDOUN AL-ZOUBI* AND MARIAM AL-AZAIZEH
Abstract.
Let G be a group with identity e . Let R be a G -graded commutative ring and M a graded R -module. In this paper,we introduce the concept of graded classical and graded stronglyclassical 2-absorbing second submodules of graded modules over agraded commutative rings. A number of results concerning of theseclasses of graded submodules and their homogeneous componentsare given. Introduction
Badawi in [19] introduced the concept of 2-absorbing ideals of com-mutative rings. Later on, Anderson and Badawi in [13] generalizedthe concept of 2-absorbing ideals of commutative rings to the conceptof n -absorbing ideals of commutative rings for every positive integer n ≥ . In light of [19] and [13], many authors studied the concept of2-absorbing submodules and n -absorbing submodules. In [17], the au-thors introduced and studied the concepts of 2-absorbing and strongly2-absorbing second submodules. In [23], the authors introduced andstudied the concept of classical 2-absorbing submodules as a general-ization of 2-absorbing submodules. Recently, H. Ansari-Toroghy andF. Farshadifar in [14] introduced and studied the concepts of classicaland strongly classical 2-absorbing second submodules of modules overcommutative rings.The scope of this paper is devoted to the theory of graded modulesover graded commutative rings. One use of rings and modules withgradings is in describing certain topics in algebraic geometry. Here,in particular, we are dealing with graded classical and graded strongly Mathematics Subject Classification.
Key words and phrases. graded classical 2-absorbing second submodules, gradedstrongly classical 2-absorbing second submodules, graded 2-absorbing second sub-modules. ∗ Corresponding author. classical 2-absorbing second submodules of graded modules over gradedcommutative rings.Graded prime ideals of graded commutative rings have been intro-duced and studied in [10, 30-32]. S.E. Atani in [18] extended gradedprime ideals to graded prime submodules. Several authors investigatedproperties of graded prime submodules, for examples see [1, 9, 11, 29].The concept of graded 2-absorbing ideals, generalizations of gradedprime ideals, were studied by K. Al-Zoubi and R. Abu-Dawwas, andother authors, (see [3, 25].) K. Al-Zoubi and R. Abu-Dawwas in [2]extended graded 2-absorbing ideals to graded 2-absorbing submodules.Later on, M. Hamoda and A. E. Ashour in [24] introduced the con-cept of graded n-absorbing submodules that is a generalization of theconcept of graded prime ideals. The notion of graded classical primesubmodules as a generalization of graded prime submodules was in-troduced in [21] and studied in[9, 6, 8, 12]. The notion of gradedsecond submodules was introduced in [15] and studied in [7, 16, 20].In[4], the authors introduced and studied the concept of graded classi-cal 2-absorbing submodules as a generalization of graded 2-absorbingsubmodules.Recently, K. Al-Zoubi and M. Al-Azaizeh in [5] introduced and stud-ied the concepts of graded 2-absorbing and graded strongly 2-absorbingsecond submodules.Here, we introduce the concept of graded classical (resp. gradedstrongly classical) 2-absorbing second submodules of graded modulesover a commutative graded rings as a generalization of graded 2-absorbing(resp. graded strongly 2-absorbing) second submodules and investigatesome properties of these classes of graded submodules.2.
Preliminaries
Convention . Throughout this paper all rings are commutative withidentity and all modules are unitary. First, we recall some basic prop-erties of graded rings and modules which will be used in the sequel. Werefer to [22], [26], [27] and [28] for these basic properties and more infor-mation on graded rings and modules. Let G be a multiplicative groupand e denote the identity element of G . A ring R is called a graded ring(or G -graded ring) if there exist additive subgroups R α of R indexed bythe elements α ∈ G such that R = L α ∈ G R α and R α R β ⊆ R αβ for all α , β ∈ G . The elements of R α are called homogeneous of degree α and allthe homogeneous elements are denoted by h ( R ), i.e. h ( R ) = ∪ α ∈ G R α .If r ∈ R , then r can be written uniquely as P α ∈ G r α , where r α is calleda homogeneous component of r in R α . Moreover, R e is a subring of N GRADED CLASSICAL 2-ABSORBING SECOND SUBMODULES 3 R and 1 ∈ R e . Let R = L α ∈ G R α be a G -graded ring. An ideal I of R is said to be a graded ideal if I = L α ∈ G ( I ∩ R α ) := L α ∈ G I α .Let R = L α ∈ G R α be a G -graded ring. A Left R -module M is said tobe a graded R -module (or G -graded R -module ) if there exists a familyof additive subgroups { M α } α ∈ G of M such that M = L α ∈ G M α and R α M β ⊆ M αβ for all α, β ∈ G . Here, R α M β denotes the additive sub-group of M consisting of all finite sums of elements r α m β with r α ∈ R α and m β ∈ M β . Also if an element of M belongs to ∪ α ∈ G M α = h ( M ),then it is called a homogeneous. Note that M α is an R e -module forevery α ∈ G . So, if I = L α ∈ G I α is a graded ideal of R , then I α isan R e -module for every α ∈ G . Let R = L α ∈ G R α be a G -gradedring. A submodule N of M is said to be a graded submodule of M if N = L α ∈ G ( N ∩ M α ) := L α ∈ G N α . In this case, N α is called the α -component of N .Let R be a G -graded ring and M a graded R -module. A propergraded submodule C of M is said to be a completely graded irreducible if C = ∩ α ∈ ∆ C α , where { C α } α ∈ ∆ is a family of graded submodulesof M , implies that C = C β for some β ∈ ∆ (see [5].) A non-zerograded submodule S of M is said to be a graded 2-absorbing secondsubmodule of M if whenever r, t ∈ h ( R ) , C is a completely gradedirreducible submodule of M , and rtS ⊆ C, then rS ⊆ C or tS ⊆ C or rt ∈ Ann R ( S ) (see [5].) A non-zero graded submodule S of M is said tobe a graded strongly 2-absorbing second submodule of M if whenever r, t ∈ h ( R ) , C , C are completely graded irreducible submodules of M , and rtS ⊆ C ∩ C , then rS ⊆ C ∩ C or tS ⊆ C ∩ C or rt ∈ Ann R ( S ) (see [5].) A proper graded submodule C of M is saidto be a graded classical 2-absorbing submodule of M if C = M ; andwhenever r, s, t ∈ h ( R ) and m ∈ h ( M ) with rstm ∈ C , then either rsm ∈ C or rtm ∈ C or stm ∈ C (see [4].)3. Graded classical 2-absorbing second submodules
Definition 3.1.
Let R be a G -graded ring and M a graded R -module.A non-zero graded submodule C of M is said to be a graded classical2-absorbing second submodule of M, if whenever r α , s β , t γ ∈ h ( R ), U is a completely graded irreducible submodule of M and r α s β t γ C ⊆ U ,then either r α s β C ⊆ U or s β t γ C ⊆ U or r α t γ C ⊆ U . We say that M is a graded classical 2-absorbing second module if M is a graded classical2-absorbing second submodule of itself. Lemma 3.2.
Let R be a G -graded ring, M a graded R -module and C agraded classical 2-absorbing second submodule of M . Let I = L γ ∈ G I γ be a graded ideal of R . Then for every r α , s β ∈ h ( R ) , γ ∈ G and K. AL-ZOUBI AND MARIAM AL-AZAIZEH completely graded irreducible submodule U of M with r α s β I γ C ⊆ U ,either r α I γ C ⊆ U or s β I γ C ⊆ U or r α s β C ⊆ U .Proof. Let r α , s β ∈ h ( R ), γ ∈ G and U be a completely graded ir-reducible submodule of M such that r α s β I γ C ⊆ U , r α I γ C * U and s β I γ C * U. We have to show that r α s β C ⊆ U. By our assumptionthere exist i γ , i ′ γ ∈ I γ such that r α i γ C * U and s β i ′ γ C * U. As r α s β i γ C ⊆ U and C is a graded classical 2-absorbing second sub-module, we have either r α s β C ⊆ U or s β i γ C ⊆ U. Similarly, by r α s β i ′ γ C ⊆ U, we get either r α s β C ⊆ U or r α i ′ γ C ⊆ U. If r α s β C ⊆ U, we are done. Suppose that r α s β C * U, which implies that s β i γ C ⊆ U and r α i ′ γ C ⊆ U. By ( i γ + i ′ γ ) ∈ I γ , it follows that r α s β ( i γ + i ′ γ ) C ⊆ U. Then either r α ( i γ + i ′ γ ) C ⊆ U or s β ( i γ + i ′ γ ) C ⊆ U as C is a gradedclassical 2-absorbing second submodule of M. If r α ( i γ + i ′ γ ) C ⊆ U, then r α i γ C ⊆ U since r α i ′ γ C ⊆ U, which is a contradiction. Similarly, if s β ( i γ + i ′ γ ) C ⊆ U, we get a contradiction. Thus r α s β C ⊆ U. (cid:3) Lemma 3.3.
Let R be a G -graded ring, M a graded R -module and C agraded classical 2-absorbing second submodule of M . Let I = L β ∈ G I β and J = L γ ∈ G J γ be a graded ideals of R. Then for every r α ∈ h ( R ) , β, γ ∈ G and completely graded irreducible submodule U of M with r α I β J γ C ⊆ U , either r α I β C ⊆ U or r α J γ C ⊆ U or I β J γ C ⊆ U. Proof.
Let r α ∈ h ( R ), β, γ ∈ G and U be a completely graded irre-ducible submodule of M such that r α I β J γ C ⊆ U, r α I β C * U and r α J γ C * U. We have to show that I β J γ C ⊆ U. Let i β ∈ I β and j γ ∈ J γ . By our assumption, there exist i ′ β ∈ I β and j ′ γ ∈ J γ such that r α i ′ β C * U and r α j ′ γ C * U. Since r α i ′ β J γ C ⊆ U, r α i ′ β C * U and r α J γ C * U, byLemma 3.2, we get i ′ β J γ C ⊆ U. Similarly, by r α j ′ γ I β C ⊆ U, r α j ′ γ C * U and r α I β C * U, we get j ′ γ I β C ⊆ U. By ( i β + i ′ β ) ∈ I β and ( j γ + j ′ γ ) ∈ J γ , it follow that r α ( i β + i ′ β )( j γ + j ′ γ ) C ⊆ U. Since C is a graded classical2-absorbing second submodule, we have either r α ( i β + i ′ β ) C ⊆ U or r α ( j γ + j ′ γ ) C ⊆ U or ( i β + i ′ β )( j γ + j ′ γ ) C ⊆ U. If r α ( i β + i ′ β ) C ⊆ U, then r α i β C * U since r α i ′ β C * U. Since r α i β J γ C ⊆ U, r α J γ C * U and r α i β C * U, by Lemma 3.2, we get i β J γ C ⊆ U and hence i β j γ C ⊆ U. Similarly, if r α ( j γ + j ′ γ ) C ⊆ U, we conclude that i β j γ C ⊆ U. If( i β + i ′ β )( j γ + j ′ γ ) C ⊆ U, then i β j γ C + i β j ′ γ C + i ′ β j γ C + i ′ β j ′ γ C ⊆ U ,this implies that i β j γ C ⊆ U. Thus I β J γ C ⊆ U. (cid:3) The following theorem give us a characterization of graded classical2-absorbing second submodule of a graded module.
Theorem 3.4.
Let R be a G -graded ring, M a graded R -module and C a non-zero graded submodule of M . Let I = L α ∈ G I α , J = L β ∈ G J β N GRADED CLASSICAL 2-ABSORBING SECOND SUBMODULES 5 and K = L γ ∈ G K γ be a graded ideals of R . Then the following state-ments are equivalent: (i) C is a graded classical 2-absorbing second submodule of M ; (ii) For every α, β, γ ∈ G and completely graded irreducible submodule U of M with I α J β K γ C ⊆ U , either J β K γ C ⊆ U or I α J β C ⊆ U or I α K γ C ⊆ U. Proof. ( i ) ⇒ ( ii )Assume that C is a graded classical 2-absorbing secondsubmodule of M. Let U be a completely graded irreducible submoduleof M and α, β, γ ∈ G such that I α J β K γ C ⊆ U and J β K γ C * U. Thenby Lemma 3.3, for all i α ∈ I α , we have either i α J β C ⊆ U or i α K γ C ⊆ U. If i α J β C ⊆ U for all i α ∈ I α , then I α J β C ⊆ U, we are done. Similarly,if i α K γ C ⊆ U for all i α ∈ I α , then I α K γ C ⊆ U, we are done. Supposethat there exist i α , i ′ α ∈ I α such that i α J β C * U and i ′ α K γ C * U. Since i α J β K γ C ⊆ U, i α J β C * U and J β K γ C * U, by Lemma 3.3, we get i α K γ C ⊆ U. Similarly, by i ′ α J β K γ C ⊆ U, i ′ α K γ C * U and J β K γ C * U, we get i ′ α J β C ⊆ U. Since ( i α + i ′ α ) J β K γ C ⊆ U and J β K γ C * U , byLemma 3.3, we get either ( i α + i ′ α ) J β C ⊆ U or ( i α + i ′ α ) K γ C ⊆ U. If ( i α + i ′ α ) J β C ⊆ U, then i ′ α J β C * U since i α J β C * U, which is acontradiction. Similarly, if ( i α + i ′ α ) K γ C ⊆ U, we get a contradiction.Therefore either I α J β C ⊆ U or I α K γ C ⊆ U. ( ii ) ⇒ ( i ) Assume that ( ii ) holds. Let r α , s β , t γ ∈ h ( R ) and U be acompletely graded irreducible submodule of M such that r α s β t γ C ⊆ U. Let I = r α R, J = s β R and K = t γ R be a graded ideals of R generatedby r α , s β and t γ , respectively. Then I α J β K γ C ⊆ U. By our assumption,we have either J β K γ C ⊆ U or I α J β C ⊆ U or I α K γ C ⊆ U. This yieldsthat either s β t γ C ⊆ U or r α s β C ⊆ U or r α t γ C ⊆ U. Thus C is a gradedclassical 2-absorbing second submodule of M . (cid:3) Corollary 3.5.
Let R be a G -graded ring, M a graded R -module and C a graded classical 2-absorbing second submodule of M and I = L α ∈ G I α be a graded ideal of R. Then for each α ∈ G, I nα C = I n +1 α C for all n ≥ . Proof.
It is enough to show that I α C = I α C. It is clear that I α C ⊆ I α C. Let U be a completely graded irreducible submodule of M suchthat I α C ⊆ U. By Theorem 3.4, we get I α C ⊆ U . This yields that I α C ⊆ I α C by [5, Lemma 2.3]. Therefore I α C = I α C. (cid:3) Clearly every graded 2-absorbing second submodule is a graded clas-sical 2-absorbing second submodule. The following example shows thatthe converse is not true in general.
Example 3.6.
Let G = Z , then R = Z is a G -graded ring with R = Z and R = { } . Let M = Z be a graded R -module with M = Z and K. AL-ZOUBI AND MARIAM AL-AZAIZEH M = { } . Now, consider the graded submodule N = 2 Z . Then N isa graded classical 2-absorbing second submodule of R which is not agraded 2-absorbing second submodule of M. Let R be a G -graded ring, M a graded R -module and N a gradedsubmodule of M . Then ( N : R M ) is defined as ( N : R M ) = { r ∈ R | rM ⊆ N } . It is shown in [18, Lemma 2.1] that if N is a gradedsubmodule of M , then ( N : R M ) = { r ∈ R : rM ⊆ N } is a gradedideal of R . The annihilator of M is defined as (0 : R M ) and is denotedby Ann R ( M ) . Recall that a proper graded ideal I of a G-graded ring R is said to be a graded 2-absorbing ideal of R if whenever r, s, t ∈ h ( R )with rst ∈ I , then rs ∈ I or rt ∈ I or st ∈ I (see [3].) Theorem 3.7.
Let R be a G -graded ring, M a graded R -module and C a graded classical 2-absorbing second submodule of M . Then for eachcompletely graded irreducible submodule U of M with C * U, ( U : R C ) is a graded 2-absorbing ideal of R. Proof.
Let r α , s β , t γ ∈ h ( R ) such that r α s β t γ ∈ ( U : R C ) . Then r α s β t γ C ⊆ U. Since C is a graded classical 2-absorbing second submod-ule, we have either s β t γ C ⊆ U or r α s β C ⊆ U or r α t γ C ⊆ U, it followsthat either s β t γ ∈ ( U : R C ) or r α s β ∈ ( U : R C ) or r α t γ ∈ ( U : R C ) . Thus ( U : R C ) is a graded 2-absorbing ideal of R. (cid:3) Theorem 3.8.
Let R be a G -graded ring, M a graded R -module and C a graded classical 2-absorbing second submodule of M . Let I = L λ ∈ G I λ be a graded ideal of R and λ ∈ G with I λ * Ann R ( C ) . Then I λ C is a graded classical 2-absorbing second submodule of M. Proof.
Since I λ * Ann R ( C ) , I λ C = 0 . Now, let r α , s β , t γ ∈ h ( R )and U be a completely graded irreducible submodule of M such that r α s β t γ I λ C ⊆ U. By Lemma 3.2, we have either r α s β I λ C ⊆ U or t γ I λ C ⊆ U or r α s β t γ C ⊆ U. If r α s β I λ C ⊆ U or t γ I λ C ⊆ U, then weare done. If r α s β t γ C ⊆ U, then either s β t γ C ⊆ U or r α t γ C ⊆ U or r α s β C ⊆ U as C is a graded classical 2-absorbing second submodule, itfollows that either s β t γ I λ C ⊆ U or r α t γ I λ C ⊆ U or r α s β I λ C ⊆ U. Thus I λ C is a graded classical 2-absorbing second submodule of M. (cid:3) Let R be a G -graded ring and M , M ′ graded R -modules. Let f : M → M ′ be an R -module homomorphism. Then f is said tobe a graded homomorphism if f ( M α ) ⊆ M ′ α for all α ∈ G , (see [28].)The category of graded R -modules possesses direct sums, products in-jective and projective limits. A graded homomorphism that is an in-jective function will be referred to simply as a monomorphism and agraded homomorphism that is a surjective function will be called anepimorphism. N GRADED CLASSICAL 2-ABSORBING SECOND SUBMODULES 7
Theorem 3.9.
Let R be a G -graded ring and M , M be two graded R -modules. Let f : M → M be a graded monomorphism. Then wehave the following: (i) If C is a graded classical 2-absorbing second submodule of M ,then f ( C ) is a graded classical 2-absorbing second submodule of f ( M ) . (ii) If C is a graded classical 2-absorbing second submodule of f ( M ) ,then f − ( C ) is a graded classical 2-absorbing second submoduleof M . Proof. ( i ) Assume that C is a graded classical 2-absorbing second sub-module of M . It is easy to see f ( C ) = 0 . Now, Let r α , s β , t γ ∈ h ( R )and U be a completely graded irreducible submodule of f ( M ) suchthat r α s β t γ f ( C ) ⊆ U . Consequently r α s β t γ C ⊆ f − ( U ) . By [5,Lemma 2.11(ii)], we have f − ( U ) is a completely graded irreduciblesubmodule of M . Then either r α s β C ⊆ f − ( U ) or s β t γ C ⊆ f − ( U )or r α s β C ⊆ f − ( U ) as C is a graded classical 2-absorbing second sub-module of M . If r α s β C ⊆ f − ( U ) , then r α s β f ( C ) = f ( r α s β C ) ⊆ f ( f − ( U )) = U ∩ f ( M ) = U . Similarly, if s β t γ C ⊆ f − ( U ) , we get s β t γ f ( C ) ⊆ U , also if r α s β C ⊆ f − ( U ), we get r α s β f ( C ) ⊆ U . Thus f ( C ) is a graded classical 2-absorbing second submodule of f ( M ) . ( ii ) Assume that C is a graded classical 2-absorbing second submod-ule of f ( M ) . It is easy to see f − ( C ) = 0 . Now, let r α , s β , t γ ∈ h ( R )and U be a completely graded irreducible submodule of M such that r α s β t γ f − ( C ) ⊆ U . Consequently r α s β t γ C = r α s β t γ ( f ( M ) ∩ C ) = r α s β t γ f f − ( C ) = f ( r α s β t γ f − ( C )) ⊆ f ( U ) . By [5, Lemma 2.11(i)],we have f ( U ) is a completely graded irreducible submodule of f ( M ) . Then either r α s β C ⊆ f ( U ) or s β t γ C ⊆ f ( U ) or r α t γ C ⊆ f ( U )as C is a graded classical 2-absorbing second submodule of f ( M ) . This yields that either r α s β f − ( C ) ⊆ U or s β t γ f − ( C ) ⊆ U or r α t γ f − ( C ) ⊆ U . Thus f − ( C ) is a graded classical 2-absorbing sec-ond submodule of M . (cid:3) Graded strongly classical 2-absorbing secondsubmodules
Definition 4.1.
Let R be a G -graded ring and M a graded R -module.A non-zero graded submodule C of M is said to be a graded stronglyclassical 2-absorbing second submodule of M if whenever r α , s β , t γ ∈ h ( R ), U , U and U are completely graded irreducible submodulesof M, and r α s β t γ C ⊆ U ∩ U ∩ U , then r α s β C ⊆ U ∩ U ∩ U or s β t γ C ⊆ U ∩ U ∩ U or r α t γ C ⊆ U ∩ U ∩ U . We say that M is a K. AL-ZOUBI AND MARIAM AL-AZAIZEH graded strongly classical 2-absorbing second module if M is a gradedstrongly classical 2-absorbing second submodule of itself.The following results give us a characterization of graded stronglyclassical 2-absorbing second submodule of a graded module. Theorem 4.2.
Let R be a G -graded ring, M a graded R -module and C a non-zero graded submodule of M . Then the following statementsare equivalent: (i) C is a graded strongly classical 2-absorbing second submodule of M ; (ii) For every r α , s β , t γ ∈ h ( R ) and every graded submodule N of M with r α s β t γ C ⊆ N, then either r α s β C ⊆ N or s β t γ C ⊆ N or r α t γ C ⊆ N ;(iii) For every r α , s β , t γ ∈ h ( R ) , either r α s β t γ C = r α s β C or r α s β t γ C = s β t γ C or r α s β t γ C = r α t γ C. Proof. ( i ) ⇒ ( ii ) Assume that C is a graded strongly classical 2-absorbing second submodule of M. Let r α , s β , t γ ∈ h ( R ) and N be agraded submodule of M such that r α s β t γ C ⊆ N. Assume on the con-trary that r α s β C * N , s β t γ C * N and r α t γ C * N. Then there existcompletely graded irreducible submodules U , U and U of M suchthat N ⊆ U , N ⊆ U and N ⊆ U but r α s β C * U , s β t γ C * U and r α t γ C * U by [5, Lemma 2.3]. Hence r α s β t γ C ⊆ U ∩ U ∩ U . Since C is a graded strongly classical 2-absorbing second submodule of M, weget either r α s β C ⊆ U ∩ U ∩ U or s β t γ C ⊆ U ∩ U ∩ U or r α t γ C ⊆ U ∩ U ∩ U which are contradiction.( ii ) ⇒ ( iii ) Assume ( ii ) is hold. Let r α , s β , t γ ∈ h ( R ). Then r α s β t γ C is a graded submodule of M . Since r α s β t γ C ⊆ r α s β t γ C, by ( ii ) , we haveeither r α s β C ⊆ r α s β t γ C or s β t γ C ⊆ r α s β t γ C or r α t γ C ⊆ r α s β t γ C. Thisyields that either r α s β t γ C = r α s β C or r α s β t γ C = s β t γ C or r α s β t γ C = r α t γ C . ( iii ) ⇒ ( i ) Trivial. (cid:3) Lemma 4.3.
Let R be a G -graded ring, M a graded R -module and C agraded strongly classical 2-absorbing second submodule of M . Let I = L γ ∈ G I γ be a graded ideal of R . Then for every r α , s β ∈ h ( R ) , γ ∈ G and graded submodule N of M with r α s β I γ C ⊆ N , either r α I γ C ⊆ N or s β I γ C ⊆ N or r α s β C ⊆ N .Proof. By using Theorem 4.2 the proof is similar to the poof of Lemma3.2, so we omit it. (cid:3)
Lemma 4.4.
Let R be a G -graded ring, M a graded R -module and C a graded strongly classical 2-absorbing second submodule of M . Let N GRADED CLASSICAL 2-ABSORBING SECOND SUBMODULES 9 I = L β ∈ G I β and J = L γ ∈ G J γ be a graded ideals of R. Then for every r α ∈ h ( R ) , β, γ ∈ G and graded submodule N of M with r α I β J γ C ⊆ N ,either r α I β C ⊆ N or r α J γ C ⊆ N or I β J γ C ⊆ N. Proof.
By using Theorem 4.2 and Lemma 4.3, the proof is similar tothe poof of Lemma 3.3, so we omit it. (cid:3)
Theorem 4.5.
Let R be a G -graded ring, M a graded R -module and C a non-zero graded submodule of M . Let I = L α ∈ G I α , J = L β ∈ G J β and K = L γ ∈ G K γ be a graded ideals of R . Then the following state-ments are equivalent: (i) C is a graded strongly classical 2-absorbing second submodule of M . (ii) For every α, β, γ ∈ G and graded submodule N of M with I α J β K γ C ⊆ N , either J β K γ C ⊆ N or I α J β C ⊆ N or I α K γ C ⊆ N. Proof.
By using Theorem 4.2 and Lemma 4.4, the proof is similar tothe proof of Theorem 3.4, so we omit it. (cid:3)
Theorem 4.6.
Let R be a G -graded ring, M a graded R -module and C a non-zero graded submodule of M and N a graded submodule of M with C * N. Then C is a graded strongly classical 2-absorbing secondsubmodule of M if and only if ( N : R C ) is a graded 2-absorbing idealof R. Proof.
Assume that C is a graded strongly classical 2-absorbing secondsubmodule of M. Since C * N, ( N : R C ) = R. Now let r α , s β , t γ ∈ h ( R )such that r α s β t γ ∈ ( N : R C ) . Then r α s β t γ C ⊆ N. By Theorem 4.2,we have either r α s β C ⊆ N or s β t γ C ⊆ N or r α t γ C ⊆ N. Thus either r α s β ∈ ( N : R C ) or s β t γ ∈ ( N : R C ) or r α t γ ∈ ( N : R C ) . Therefore( N : R C ) is a graded 2-absorbing ideal of R. Conversely, Let r α , s β ,t γ ∈ h ( R ) and N be a graded submodule of M with r α s β t γ C ⊆ N. If C ⊆ N, then we are done. Assume that C * N. By our assumption, wehave ( N : R C ) is a graded 2-absorbing ideal of R. Since r α s β t γ ∈ ( N : R C ), we conclude that either r α s β C ⊆ N or s β t γ C ⊆ N or r α t γ C ⊆ N .Hence C is a graded strongly classical 2-absorbing second submoduleof M by Theorem 4.2. (cid:3) Clearly every graded strongly 2-absorbing second submodule is agraded strongly classical 2-absorbing second submodule. The followingexample shows that the converse is not true in general.
Example 4.7.
Let G = Z , then R = Z is a G -graded ring with R = Z and R = { } . Let M = Z × Q be a graded R -module with M = Z × Q and M = { (0 , } . Then M is a graded strongly classical2-absorbing second module which is not a graded strongly 2-absorbingsecond module. Theorem 4.8.
Let R be a G -graded ring and M , M be two graded R -modules. Let f : M → M be a graded monomorphism. Then wehave the following. (i) If C is a graded strongly classical 2-absorbing second submoduleof M , then f ( C ) is a graded strongly classical 2-absorbing secondsubmodule of M . (ii) If C is a graded strongly classical 2-absorbing second submoduleof f ( M ) , then f − ( C ) is a graded strongly classical 2-absorbingsecond submodule of M . Proof. ( i ) Assume that C is a graded strongly classical 2-absorbing sec-ond submodule of M. It is easy to see f ( C ) = 0 . Let r α , s β , t γ ∈ h ( R ) . By Theorem 4.2, we have either r α s β t γ C = r α s β C or r α s β t γ C = s β t γ C or r α s β t γ C = r α t γ C . We can assume that r α s β t γ C = r α s β C . Then r α s β t γ f ( C ) = f ( r α s β t γ C ) = f ( r α s β C ) = r α s β f ( C ) . Hence f ( C ) is a graded strongly classical 2-absorbing second submodule of M by Theorem 4.2.( ii ) Use the technique of Theorem 3.9(ii), and apply Theorem 4.2. (cid:3) Definition 4.9.
Let S be a non-zero graded submodule of a graded R -module M . We say that S is a graded weakly second submodule of M if r α s β S ⊆ N, where r α , s β ∈ h ( R ) and N is a graded submodule of M , then either r α S ⊆ N or s β S ⊆ N. Lemma 4.10.
Let R be a G -graded ring and M be a graded R -module.If S and S are graded weakly second submodules of M, then S = S + S is a graded strongly classical 2-absorbing second submodule of M. Proof.
Assume that S and S are graded weakly second submodules of M and S = S + S . Let r α , s β , t γ ∈ h ( R ). As r α s β t γ S ⊆ r α s β t γ S and S is a graded weakly second submodule, then either r α S ⊆ r α s β t γ S or s β S ⊆ r α s β t γ S or t γ S ⊆ r α s β t γ S . This yields that either r α S = r α s β t γ S or s β S = r α s β t γ S or t γ S = r α s β t γ S . Similarly, we haveeither r α S = r α s β t γ S or s β S = r α s β t γ S or t γ S = r α s β t γ S . We mayassume that r α S = r α s β t γ S . Likewise assume that s β S = r α s β t γ S . Then r α s β t γ S = r α s β t γ S + r α s β t γ S = r α S + s β S = r α s β S = r α s β S + r α s β S = r α s β S. By Theorem 4.2, we have S is a graded stronglyclassical 2-absorbing second submodules of M . (cid:3) N GRADED CLASSICAL 2-ABSORBING SECOND SUBMODULES 11
Let R i be a graded commutative ring with identity and M i be agraded R i -module, for i = 1 ,
2. Let R = R × R . Then M = M × M is a graded R -module and each graded submodule C of M is of theform C = C × C for some graded submodules C of M and C of M (see [28].) Theorem 4.11.
Let R = R × R be a G -graded ring and M = M × M a graded R -module where M is a graded R -module and M is a graded R -module. Let C and C be a non-zero graded submodules of M and M , respectively. (i) C = C × is a graded strongly classical 2-absorbing second sub-module of M if and only if C is a graded strongly classical 2-absorbing second submodule of M . (ii) C = 0 × C is a graded strongly classical 2-absorbing second sub-module of M if and only if C is a graded strongly classical 2-absorbing second submodule of M . (iii) C = C × C is a graded strongly classical 2-absorbing submoduleof M if and only if C and C are graded weakly second submodulesof M and M , respectively.Proof. ( i ) Assume that C = C × M . From our assumption, C is a non-zero graded submodule, so C = 0 . Set M ′ = M × . One can see that C = C × M ′ . Also, observe that M ′ ≃ M and C ≃ C . Hence C is a gradedstrongly classical 2-absorbing second submodule of M by Theorem 4.8.The converse is clear.( ii ) It can be easily verified similar to ( i ).( iii ) Assume that C = C × C is a graded strongly classical 2-absorbing submodule of M. We have to show that C is a graded weaklysecond submodule of M . Since C = 0 , by [5, Lemma 2.3], there exist acompletely graded irreducible submodule U of M such that C * U . Let r α , s β ∈ h ( R ), N be a graded submodule of M and r α s β C ⊆ N. Thus ( r α , s β , , C × C ) = r α s β C × ⊆ N × U . As C = C × C is a graded strongly classical 2-absorbing submodule of M, by Theorem4.2, we have either ( r α , s β , C × C ) = r α s β C × C ⊆ N × U or( s β , , C × C ) = s β C × ⊆ N × U or ( r α , , C × C ) = r α C × ⊆ N × U . If r α s β C × C ⊆ N × U , then C ⊆ U a contradiction. Whichimplies either r α C ⊆ N or s β C ⊆ N. Thus C is a graded weaklysecond submodule of M . Similarly, we can show that C is a gradedweakly second submodule of M . Conversely, assume that C and C are graded weakly second sub-modules of M and M , respectively. Then it is clear that C × , × C are graded weakly second submodules of M. By Lemma 4.10,we have C = C × × C is a graded strongly classical 2-absorbingsubmodule of M . (cid:3) Acknowledgments
The authors wish to thank sincerely the referees for their valuable com-ments and suggestions.
Conflict of Interest Statement:
On behalf of all authors, thecorresponding author states that there is no conflict of interest.
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On the union of graded prime ideals ,Open Phys., (2016), 114–118. Khaldoun Al-Zoubi, Department of Mathematics and Statistics, Jordan Univer-sity of Science and Technology, P.O.Box 3030, Irbid 22110, Jordan.Email address : [email protected] Mariam Al-Azaizeh, Department of Mathematics, University of Jordan, Am-man, Jordan..Email address ::