On graded quasi-semiprime submodules of graded modules over graded commutative rings
aa r X i v : . [ m a t h . A C ] J a n ON GRADED QUASI-SEMIPRIME SUBMODULES OFGRADED MODULES OVER GRADED COMMUTATIVERINGS
KHALDOUN AL-ZOUBI* AND SHATHA ALGHUEIRI
Abstract.
Let G be a group with identity e . Let R be a G -graded commutative ring and M a graded R -module. A propergraded submodule N of M is called a graded semiprime submodule if whenever r ∈ h ( R ), m ∈ h ( M ) and n ∈ Z + with r n m ∈ N , then rm ∈ N . In this paper, we introduce the concept of graded quasi-semiprime submodule as a generalization of graded semiprime sub-module and show a number of results in this class. We say thata proper graded submodule N of M is a graded quasi-semiprimesubmodule if ( N : R M ) is a graded semiprime ideal of R . Introduction and Preliminaries
Throughout this paper all rings are commutative with identity andall modules are unitary.Graded semiprime submodules of graded modules over graded com-mutative rings, have been introduced and studied in [1, 5, 7, 12]. Also,the concept of graded semiprime ideal was introduced by Lee and Var-mazyar [7] and studied in [4].Recently, K. Al-Zoubi, R. Abu-Dawwas and I. Al-Ayyoub in [1] in-troduced and studied the concept of graded semi-radical of graded sub-modules in graded modules.Here, we introduce the concept of graded quasi-semiprime submod-ules of graded modules over a commutative graded rings as a generaliza-tion of graded semiprime submodules and investigate some propertiesof these classes of graded submodules.First, we recall some basic properties of graded rings and moduleswhich will be used in the sequel. We refer to [6] and [8-10] for thesebasic properties and more information on graded rings and modules.
Mathematics Subject Classification.
Key words and phrases. graded quasi-semiprime submodule, graded semiprimesubmodule, graded prime. ∗ Corresponding author.
Let G be a multiplicative group and e denote the identity elementof G . A ring R is called a graded ring (or G -graded ring) if thereexist additive subgroups R α of R indexed by the elements α ∈ G suchthat R = L α ∈ G R α and R α R β ⊆ R αβ for all α , β ∈ G . The elementsof R α are called homogeneous of degree α and all the homogeneouselements 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 called a homogeneouscomponent of r in R α . Moreover, R e is a subring of 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 agraded ideal if I = L α ∈ G ( I ∩ R α ) := L α ∈ G I α . Let R = L α ∈ G R α bea G -graded ring. A Left R -module M is said to be a graded R -module (or G -graded R -module ) if there exists a family of 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 subgroup of M consistingof all finite sums of elements r α m β with r α ∈ R α and m β ∈ M β . Alsoif an element of M belongs to ∪ α ∈ G M α = h ( M ), then it is called ahomogeneous. Note that M α is an R e -module for every α ∈ G . So, if I = L α ∈ G I α is a graded ideal of R , then I α is an R e -module for every α ∈ G . Let R = L α ∈ G R α be a G -graded ring. 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, 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 [3, Lemma 2.1] that if N is a graded submod-ule of M , then ( N : R M ) is a graded ideal of R . The annihilator of M is defined as (0 : R M ) and is denoted by Ann R ( M ).A proper graded submodule N of M is said to be a graded semiprimesubmodule if whenever r ∈ h ( R ), m ∈ h ( M ) and n ∈ Z + with r n m ∈ N , then rm ∈ N , (see [5].) A proper graded ideal I of R is said tobe graded semiprime ideal if whenever r, s ∈ h ( R ) and n ∈ Z + with r n s ∈ I , then rs ∈ I , (see [4].)2. Results
Definition 2.1.
Let R be a G -graded ring and M a graded R -module.A proper graded submodule N of M is said to be a graded quasi-semiprime submodule of M if ( N : R M ) is a graded semiprime ideal of R . Theorem 2.2.
Let R be a G -graded ring, M a graded R -module and N a proper graded submodule of M . If N is a graded semiprime submoduleof M, then N is a graded quasi-semiprime submodule of M .Proof. By [1, Theorem 2.4]. (cid:3)
RADED QUASI-SEMIPRIME SUBMODULES 3
The next example shows that a graded quasi-semiprime submoduleis not necessarily graded semiprime submodule.
Example 2.3.
Let G = Z , R = Z be a G -graded ring with R = Z and R = { } . Let M = Z × Z be a graded R -module with M = Z × { } and M = { } × Z . Now, consider a submodule N = 4 Z × { } of M . Then it is a graded submodule and ( N : R M ) = { } is a gradedsemiprime ideal of R, and so N is a graded quasi-semiprime submoduleof R. But the graded submodule N is not graded semiprime submoduleof M, since 2 (3 , ∈ N but 2(3 , / ∈ N. Example 2.4.
Let G = Z , R = Z be a G -graded ring with R = Z and R = { } . Let M = Z be a G -graded R -module with M = Z and M = { } . Now, consider a submodule N = < > of M. Then itis a graded submodule and ( N : R M ) = 4 Z is not a graded semiprimeideal of R since 2 ∈ Z but 2 · / ∈ Z . Then N is not gradedquasi-semiprime submodule of M. Recall that a graded R -module M is called a graded multiplication iffor each graded submodule N of M , we have N = IM for some gradedideal I of R . If N is graded submodule of a graded multiplicationmodule M , then N = ( N : R M ) M . Theorem 2.5.
Let R be a G -graded ring, M a graded multiplication R -module and N a proper graded submodule of M . Then N is a gradedquasi-semiprime submodule of M if and only if N is a graded semiprimesubmodule of M .Proof. By [1, Theorem 2.5]. (cid:3)
Theorem 2.6.
Let R be a G -graded ring, M a graded multiplication R -module and N a proper graded submodule of M . Then the followingstatements are equivalent: (i) N is a graded quasi-semiprime submodule of M. (ii) If whenever I k M ⊆ N, where I is a graded ideal of R and k ∈ Z + , then IM ⊆ N .Proof. ( i ) ⇒ ( ii ) By Theorem 2.5 and [5, Proposition 2.6]( ii ) ⇒ ( i ) Let r k s ∈ ( N : R M ) where r, s ∈ h ( R ) and k ∈ Z + . So r k sM ⊆ N. Let I = ( rs ) be a graded ideal of R generated by rs . Then I k M ⊆ N. By our assumption we have IM = ( rs ) M ⊆ N. This yieldsthat rs ∈ ( N : R M ). So ( N : R M ) is a graded semiprime ideal of R. Therefore N is a graded quasi-semiprime submodule of M. (cid:3) Recall that a proper graded ideal I of a G -graded ring R is said to be a graded prime ideal if whenever r, s ∈ h ( R ) with rs ∈ I , then either K. AL-ZOUBI AND S. ALGHUEIRI r ∈ I or s ∈ I (see [11].) A proper graded ideal J of R is said to be agraded primary ideal if whenever r, s ∈ h ( R ) with rs ∈ J , then either r ∈ J or s n ∈ J for some n ∈ Z + (see [11].) Theorem 2.7.
Let R be a G -graded ring, M a graded R -module and N a graded quasi-semiprime submodule of M . If ( N : R M ) is a gradedprimary ideal of R , then ( N : R M ) is a graded prime ideal of R .Proof. Suppose that ( N : R M ) is a graded primary ideal of R . Let rs ∈ ( N : R M ) and r / ∈ ( N : R M ) . Then s ∈ Gr (( N : R M )) as( N : R M ) is a graded primary ideal of R. Hence s k ∈ ( N : R M ) forsome k ∈ Z + . Since ( N : R M ) is a graded semiprime ideal of R, wehave s ∈ ( N : R M ). Therefore ( N : R M ) is a graded prime ideal of R. (cid:3) Let R be a G -graded ring, M a graded R -module and N a gradedsubmodule of M . The graded envelope submodule RGE M ( N ) of N in M is a graded submodule of M generated by the set GE M ( N ) = { rm : r ∈ h ( R ) , m ∈ h ( M ) such that r n m ∈ N for some n ∈ Z + } (see [2,Definition 1].) Theorem 2.8.
Let R be a G -graded ring, M a graded multiplication R -module and N a proper graded submodule of M . Then N is a gradedquasi-semiprime submodule of M if and only if N = RGE M ( N ) . Proof.
Suppose that N is a graded quasi-semiprime submodule of M .Then N is a graded semiprime submodule of M by Theorem 2.5.Clearly, N ⊆ RGE M ( N ). Now, let x ∈ GE M ( N ) . Then x = rm for some r ∈ h ( R ) , m ∈ h ( M ) and there exists k ∈ Z + such that r k m ∈ N. Then rm ∈ N as N is a graded semiprime submodule of M . Hence GE M ( N ) ⊆ N. This yields that
RGE M ( N ) ⊆ N. Thus N = RGE M ( N ) . Conversely, suppose that N = RGE M ( N ). Let r ∈ h ( R ), m ∈ h ( M ) and k ∈ Z + such that r k m ∈ N , so by thedefinition of the set GE M ( N ) we have rm ∈ GE M ( N ) . Then rm ∈ N as GE M ( N ) ⊆ RGE M ( N ) = N , so N is a graded semiprime submod-ule of M . Therefore N is a graded quasi-semiprime submodule of M by Theorem 2.2. (cid:3) Let R be a G -graded ring and M , M ′ be two graded R -modules. Let f : M → M ′ be an R -module homomorphism. Then f is said to be agraded homomorphism if f ( M α ) ⊆ M ′ α for all α ∈ G (see [10].) Theorem 2.9.
Let R be a G -graded ring, M , M ′ be two graded R -modules and f : M → M ′ a graded epimorphism. (i) If N is a graded quasi-semiprime submodule of M such that ker ( f ) ⊆ N , then f ( N ) is a graded quasi-semiprime submodule of M ′ . RADED QUASI-SEMIPRIME SUBMODULES 5 (ii) If N ′ is a graded quasi-semiprime submodule of M ′ , then f − ( N ′ ) is a graded quasi-semiprime submodule of M. Proof. ( i ) Suppose that N is a graded quasi-semiprime submodule of M and ker ( f ) ⊆ N. It is easy to see that f ( N ) = M ′ . Now let r k s ∈ ( f ( N ) : R M ′ ) where r, s ∈ h ( R ) and k ∈ Z + , it follows that, r k sM ′ ⊆ f ( N ) . Then r k sM ′ = r k sf ( M ) = f ( r k sM ) ⊆ f ( N ) since f is anepimorphism. This yields that r k sM ⊆ N since ker ( f ) ⊆ N, i.e., r k s ∈ ( N : R M ) . Since N is a graded quasi-semiprime submodule of M , we get rs ∈ ( N : R M ) , i.e., rsM ⊆ N. Hence f ( rsM ) = rsf ( M ) = rsM ′ ⊆ f ( N ) , i.e., rs ∈ ( f ( N ) : R M ′ ) . Therefore, f ( N ) is a gradedquasi-semiprime submodule of M ′ . ( ii ) Suppose that N ′ is a graded quasi-semiprime submodule of M ′ .It is easy to see that f − ( N ′ ) = M. Let r k s ∈ ( f − ( N ′ ) : R M )where r, s ∈ h ( R ) and k ∈ Z + , it follows that, r k sM ⊆ f − ( N ′ ) . Then r k sf ( M ) = r k sM ′ ⊆ N ′ , i.e. , r k s ∈ ( N ′ : R M ′ ) . Then rs ∈ ( N ′ : R M ′ ) as N ′ is a graded quasi-semiprime submodule of M ′ . So rsM ′ = rsf ( M ) = f ( rsM ) ⊆ N ′ . It follows that rsM ⊆ f − ( N ′ ) . So rs ∈ ( f − ( N ′ ) : R M ) . Therefore f − ( N ′ ) is a graded quasi-semiprimesubmodule of M. (cid:3) Theorem 2.10.
Let R be a G -graded ring, M a graded R -module and K a proper graded submodule of M . If N is a graded quasi-semiprimesubmodule of M with N ⊆ K and ( N : R M ) is a graded maximal idealof R , then K is a graded quasi-semiprime submodule of M .Proof. Suppose that N ⊆ K, it follows that ( N : R M ) ⊆ ( K : R M ) . By [3, Lemma 2.1], ( K : R M ) is a proper graded ideal of R . Then( N : R M ) = ( K : R M ) as ( N : R M ) is a graded maximal ideal of R .This yields that ( K : R M ) is a graded semiprime ideal of R . Therefore K is a graded quasi-semiprime submodule of M . (cid:3) Theorem 2.11.
Let R be a G -graded ring, M a graded R -module and N and K be two graded quasi-semiprime submodules of M. Then N ∩ K is a graded quasi-semiprime submodule of M .Proof. Let r k s ∈ ( N ∩ K : R M ) where r, s ∈ h ( R ) and k ∈ Z + . Thisyields that r k s ∈ ( N : R M ) ∩ ( K : R M ) . Since ( N : R M ) and ( K : R M )are graded semiprime ideals of R , we have rs ∈ ( N : R M ) ∩ ( K : R M )and so rs ∈ ( N ∩ K : R M ) . Therefore N ∩ K is a graded quasi-semiprimesubmodule of M . (cid:3) Let R be a G -graded ring and M be a graded R -module, M is calleda graded semiprime module if (0) is a graded semiprime submodule of M . K. AL-ZOUBI AND S. ALGHUEIRI
Definition 2.12.
Let R be a G -graded ring and M be a graded R -module. Then M is said to be a graded quasi-semiprime module if Ann R N is a graded semiprime ideal of R , for every non-zero gradedsubmodule N of M . Theorem 2.13.
Let R be a G -graded ring and M be a graded R -module. If M is a graded semiprime module, then M is a graded quasi-semiprime module.Proof. Suppose that M is a graded semiprime module. Then 0 is agraded semiprime submodule of M. Now, Let N be a non-zero gradedsubmodule of M and r k s ∈ Ann R N where r, s ∈ h ( R ) and k ∈ Z + . It follows that r k sN = 0 . Then rsN = 0 as 0 is a graded semiprimesubmodule of M. Hence rs ∈ Ann R N, it follows that Ann R N is agraded semiprime ideal of R. Therefore M is a graded quasi-semiprimemodule. (cid:3) References [1] K. Al-Zoubi, R. Abu-Dawwas and I. Al-Ayyoub, Graded semiprime submod-ules and graded semi-radical of graded submodules in graded modules, Ricerchemat., 66(2) (2017), 449–455.[2] S.E. Atani, F.E.K. Saraei, Graded Modules which Satisfy the Gr-Radical For-mula, Thai J. Math. 8(1) (2010), 161–170.[3] S.E. Atani, On Graded Prime Submodules, Chiang Mai J. Sci. 33(1) (2006),3–7.[4] F. Farzalipour, P. Ghiasvand, On Graded Semiprime and Graded WeaklySemiprime Ideals, Int. Electron. J. Algebra 13 (2013), 15–22.[5] F. Farzalipour, P. Ghiasvand, On Graded Semiprime Submodules, Word Acad.Sci. Eng. Technol. 68 (2012), 694–697.[6] R. Hazrat, Graded Rings and Graded Grothendieck Groups, Cambridge Uni-versity Press, Cambridge, 2016.[7] S.C Lee, R. Varmazyar, Semiprime submodules of Graded multiplication mod-ules, J. Korean Math. Soc. 49(2) (2012), 435–447.[8] C. Nastasescu and F. Van Oystaeyen, Graded and ltered rings and modules,Lecture notes in mathematics 758, Berlin-New York: Springer-Verlag, 1982.[9] C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory, Mathematical Library28, North Holand, Amsterdam, 1982.[10] Nastasescu and F. Van Oystaeyen, Methods of Graded Rings, LNM 1836.Berlin-Heidelberg: Springer-Verlag, 2004.[11] M. Refai, K. Al-Zoubi, On Graded Primary Ideals, Turk. J. Math. 28 (2004),217–229.[12] H. A.Tavallaee and M. Zolfaghari, Graded weakly semiprime submodules ofgraded multiplication modules, Lobachevskii J. Math. 34(1) (2013), 61–67.
RADED QUASI-SEMIPRIME SUBMODULES 7
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] Shatha Alghueiri, Department of Mathematics and Statistics, Jordan Universityof Science and Technology, P.O.Box 3030, Irbid 22110, JordanEmail address ::