International Journal of Algebra, Vol. 7, 2013, no. 1, 17 - 24 HIKARI Ltd, www.m-hikari.com

Multiplication Graded Modules

A. Khaksari and F. Rasti Jahromi

Department of Mathematics,Payame Noor University P.O. Box:19395-3697,Tehran, Iran [email protected] [email protected]

Copyright © 2013 A. Khaksari and F. Rasti Jahromi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let G be a multiplicative group and R be a G -graded commutative ring and M a G -graded R -module. Various properties of multiplicative ideals in a graded ring are discussed and we extend this to graded modules over graded rings. We have also discussed the set of P -primary ideals and modules of R when P is a graded multiplication prime ideals and modules.

Keywords: graded rings, graded modules, graded multiplication Modules.

1. Introduction

Let G be a group. A ring R is called G -graded ring if there exist a family

R of additive subgroup of R such that RR=⊕ g that RRgh⊂ R gh for { g }gG∈ gG∈ each gh, ∈ G. A R- module M is called R -graded Module over G if

MM=⊕ g and RM⊂ M for all gh, ∈ G. Thus each M submodule gG∈ gh gh g of M is RR= g - module. An element of a graded ringR is called homogeneous if it belongs to M g . If an element mM∈ is belongs to M g , then m is gGU∈ gGU∈ called homogeneous element and the set of all homogeneous elements of M is denoted by HM()(for a ring R is denoted byHR()). A graded submodule N of a graded R -Module M (R is a graded ring) is a submodule such that NMNN=⊕() ∩ =⊕ . Equivalently, N is graded in M if and only if gG∈∈gg gG

18 A. Khaksari and F. Rasti Jahromi

N has a homogeneous set of generators. IfRR=⊕ andRR′ =⊕ ′ are two gG∈ g gG∈ g ′ graded ring, then the mapping φ : RR→ whit φ()11RR= ′ is called graded ′ homomorphism if φ()RRgg⊂ , for all gG∈ . If MM=⊕ and MM′′=⊕ are two gradedR -modules (R is a graded gG∈ g gG∈ g ring), the mapping λ : MM→ ′ is called graded homomorphism if ′ λ()MMgg⊂ ′ for all gG∈ . A graded ideal P of a graded ring R is called gr- prime if whenever x ,yHR∈ () with xyP∈ then x ∈P or y ∈P . And a graded submodule of a graded module M over a graded ring R is called gr-prime submodule ifrx∈ N , for rHR∈ () and x ∈HM(), then x ∈N or rM⊂ N . A graded ideal m of a graded ring R is called gr-maximal if it is maximal in the lattice of graded ideals ofR. (similarly we have for R -modules). A graded ring R is called a gr-local ring if it has unique gr-maximal ideal. LetR be a graded ring and let SHR⊂ () be a multiplicatively closed subset ofR . Then the ring of fraction SR−1 −−11 is a graded ring which is called a gr-ring of fractions. Indeed, SR=⊕() SRg gG∈

−1 ⎧rrdeg( ) ⎫ −−11 where ()SRg =∈∈=⎨ ,,, r RsSg ⎬ . And SM=⊕() SMg ⎩⎭ssdeg( ) gG∈

−1 ⎧⎫mMdeg( ) where ()SMg =∈∈=⎨⎬,,, m MsSg . Consider the ring gr- ⎩⎭ssdeg( ) r homomorphism π : RSR→ −1 defined byπ()r = . And π : MSM→ −1 is 1 m called gr-homomorphism if π()m = . Let P be any gr-prime ideal of a graded 1 ring R and consider the multiplicatively closed subset SHRP= ()− . We denote −1 g the graded ring of fraction SRof R by Rp and we call it the gr-localization ofR . This is a gr-local with the unique gr-maximal ideal SP−1 which will be denoted by g PR p .Let I be a graded ideal in a graded ringR . The graded radical of I (gr- rad(I )) is defined the set of all x g ∈R such that for each gG∈ , there exists

ng ng > 0 such thatx g ∈I . A graded radical submodule N of a gradedR- moduleM (R is a graded ring) is the intersections of graded prime submodules of M such that containing N as a submodules. A submodule N of on R -module M is called multiplication ifNIM= , for some gr-ideal I of R.If each sub module of M is gr-multiplication , M is called gr-multiplication R -module.

Multiplication graded modules 19

In this paper, we study some properties of gr-multiplication submodules in a graded multiplication R-module M , when M is gr-module over gr-ring R. And give a characterization of finitely generated gr-multiplication submodules of a gr- multiplication M over a gr-ring R .

Definition 1 Let R be a graded ring over the group G and M an R-graded module. A graded submodule N of M is called graded multiplication . If KN< g then there is an gr-ideal of R such that KNI= .

Definition 2 A graded R -module M is called gr-multiplication module if every gr-submodule of M is gr-multiplication .

Definition 3 A graded ideal Q of a graded ring R is called gr-primary if QR≠ and whenever, a,bHR∈ () whit abQ∈ , then a ∈Q or bQn ∈ .If Q is gr-primary ideal of R and gr-rad ()QP= , we say that Q is gr-p-primary.

Definition 4 An gr-submodule N of graded R -module M is called gr-primary, if a ∈HR(), bHM∈ () and abN∈ ,then bN∈ or a nMN⊂ for some integer n ≥ 0 . Recall that if NK, are two gr-submodules of a graded R-module M , then (:)NK=∈{ r RrKN ⊂} is a graded ideal of R .

Lemma 1 Let I be a graded ideal in a graded ring R then I is multiplication if IJIJI∩=(:) for gr-ideal JI⊂ . Proof. Suppose that JI⊂ for some gr- ideal J of R .Then JI=∩ JIJI =(:). Hence J is gr-multiplication ideal of I . Conversely, Let I be a graded multiplication ideal in R , Let J be any graded ideal of R Then IJI∩⊂, so there is a graded ideal K of R such that IJIK∩= .Therefore KIJIJI⊆∩(( ) : ) ⊆ ( : ) , and then IJIKIJI∩= ⊂(:). On the other hand, clearly IJI(:)⊂∩ I J. Hence JI=∩ J =(:) JII.

Proposition 1 Let M be a graded R -module (R is a graded ring). Then M is gr-multiplication if for every gr-submodule N of M , NNMM=[: ]. Proof. Let M be gr-multiplication R -module, and N a gr-submodule of M , then there is an gr-ideal I of R such that NIM= , as IM⊂ N we have INM⊂ [: ] and NIMMNM=⊂[: ]. Since [:NMM ] ⊂ N, so NNMM=[: ]. Conversely it is clearly. Recall that Graded R -module M is called graded cyclic if MRx= , for some x ∈HM().

20 A. Khaksari and F. Rasti Jahromi

Theorem 1 Let M be a gr-multiplication Module over a graded local ringR . Then M is gr-multiplication if M is a graded cyclic R -module . Proof. If Mm=< > for some mHM∈ () then clearly M is gr-multiplication

R -module .Conversely, Let Mm=<∈>α α Awhere each M α is a homogeneous element (())mHMα ∈ . Since M is gr-multiplication we have ,as . Rmαα=[:] M M M MRmmMMMmM==∑ αα∑∑[:] = ([:]) α αα∈∈AA α ∈ A If , then . Since otherwise if , ∑[:]mMα = R [:]mMRα = ∀α[:]mMα ≠ R 0 α∈A then [:]mMα ⊂ J, where J is the only maximal ideal of R , and hence that is a contradiction, so for some ∑[:]mMα =⊂ RJ [:]mMRα = α ∈A 0 o α∈A therefore <>=mmMMMαα[:] = Hence M is gr-principal. If oo , then , and then ∑[:]mMα ≠ R ∑[:]mMα ⊂ J α∈A α∈A therefore , hence . MmMMJMM=⊆⊂∑ [:]α JM= M M =<0 > α∈A

Proposition 2 . If M is gr-multiplication R -module where R is a graded ring, and SHR⊂ ()) is a multiplicatively closed subset of R . Then SM−1 is a gr- multiplication SR−1 -module. Proof. Let K be a graded SR−1 -submodule of SM−1 . ThenKSN= −1 for some graded submodule of N of M . Now since M is gr-multiplication R- module, then NNMM=[: ] so SN−−11= ([:])() S N M SM − 1HenceSM−1 is a gr-multiplication SR−1 -module.

Definition 5 A graded submodule N of graded R -module M is locally gr- g principal if NR⋅ p is gr-principal for every gr-prime ideal P of R.

Proposition 3 Let R be a gr-local ring with graded maximal ideal J and M a graded R -module such that Mmmm=<12,,, ⋅⋅⋅k > , where mHM1 ∈ () for every

1≤≤ik, then Mm= for some 1≤ ik≤ . Proof. Suppose that M =<>a for some a ∈HM() and k k , then and each , so M = aa= ∑rii aaii= s aa= ∑rsii i =1 i =1

Multiplication graded modules 21

k k and if is a unit in then since a(1- ∑rxii) = 0 1-∑rsii R a = 0 i =1 i =1

aaaii==s , i0 , for all 1≤ ik≤ . and M =<>=<>0 ai , for all k k k . If is not a unit, then and so is a ik=⋅⋅⋅12,, , 1-∑rsii ∑rsii∉ J ∑rsii i =1 i =1 i =1 unit. Therefore, there is an ik∈⋅⋅⋅{}12,, , such that rsii is a unit. Otherwise since k each is not unit then , for all , hence . That rsii rxii∈ M ik=12,,⋅⋅⋅ , ∑rsii∈ M i =1 is a contradiction. So rsii is a unit for some i , then si is a unit. Hence −−11 aa==∈<>ssii a i s a i Then M =<>ai .

Theorem 2 Let Mmmm=<12,,, ⋅⋅⋅k > be a finitely generated gradedR - module over a graded ring R . Then the following are equivalent . (1)M is gr-multiplication . (2) M is locally gr-principal . k (3) , where ∑[:mMii ]= R M iiik=<⋅⋅⋅⋅⋅⋅>aa12,,, a−+ 1 , a 1 ,, a i =1 Proof. (1) → (2) By Theorem 1 . (2) → (3) Let M be a locally gr-principal. Then for graded prime ideal P of R , mm m m we have by Proposition 3 MRgg=<12,,, ⋅⋅⋅ki >=< >=< m > R , p 11 1 1 1 p for some ik∈⋅⋅⋅{}12,, , }. Hence for any gr-prime ideal P of ggg RmR[(ip ) : MR ip ] = R p, where M iiik=<⋅⋅⋅⋅⋅⋅>aa12,,, a−+ 1 , a 1 ,, a and kk then gggg. Since is finitely ([():]∑∑mMRiip== [() mRMRR ipipp : ] M i ii==11 k generated for each , There for . i ∑((mMii ) : ) = R i =1 k (3) (2) Suppose that . Then for any gr-prime of we → ∑((mMii ) : ) = R P R i =1 kkk have gg g gg. ([():∑∑∑mRip MR p ]== ([():])([():]) m i M R p m i M i R p = R p iii===111 ggg Therefore, there is ik∈⋅⋅⋅{}12,, , such that ((mRip ) : MR p ) = R p and then

22 A. Khaksari and F. Rasti Jahromi

a a MRgg⊂=<>()a R i . It follows that MRg =<>i for each gr-prime ideal pip1 p 1 P of R . Hence M is locally gr-principal. If M is a graded module over the graded ring R we define the θ g ()MxM= ∑ [():]. It is clear thatθ g ()M is a graded xM∈() ideal of R .

Proposition 4 Let M be a graded multiplication module over a graded ring R . Then (1) MM= θ g () M (2) NN= θ g () M for any graded submodule N of M . Proof. (1) Let x ∈M as M is graded multiplication R -module, then <>=x [(xMM ): ] since M=<>===∑∑ x[( xMMM ): ] ∑ [( xM ): ] Mθ g ( M ). xM∈∈ xM xM ∈ (2) suppose that N is a graded submodule of M . Then NNMM=[: ], where [:NM ] is a graded ideal of R . Hence NNMMNMMMNM==[: ] [: ]()θθgg = ().

Proposition 5 Let N and K be graded submodules of graded multiplication R- module M and SHR⊂ () be a multiplicatively closed subset of R . Then (1) θθgg()NK ()⊂ θ g ( NK ) (2) SN−−11(())(θθgg⊆ SN ()) Proof. (1) If M is a multiplication R -module and NIM= andKJM= we n defined . If and , then , where , NK= IJM x ∈M y ∈K xyrm= ∑ ii rIJi ∈ i =1 for all in=⋅⋅⋅12,, , and n ≥1. See [2]. Let a ∈∩NHM() and bK∈ ∩ HM(). It is enough to prove that n . Let where [(aa ):NbK ][( ): ]⊆ [( bNK ): ] ∑x iiyNbK∈[(a ): ][( ): ] i =1

x i ∈[(a ):N ] and ybKi ∈[( ): ], for in=12,,⋅⋅⋅ , . Then xNi ⊂ ()a and

yKi ⊂ () b, for in=⋅⋅⋅12,, , . Hence, x iiyNK⊂ ()a b and then n . Therefore . x iiybNK∈[(a ): ] ∑x iiybNK∈[(a ): ] i =1

Multiplication graded modules 23

(2) SNS−−11(())θ g ⊆=⊆ (∑∑ [():] xN sxN − 1 [():] xN∈∩ HM() xN ∈∩ HM () x ∑ [( ) :sN−−11 ]⊆ θ g ( sN ). xN∈∩ HM()1

Recall that a graded module M over graded ring R is called gr-finitely generated if M is generated by a finite set of homogeneous elements .

Theorem 3 Let M be a graded R -module where R is a graded ring. Then M is gr-finitely generated and locally gr-principal if θ g ()MR= . gg Proof. Let J be a gr-maximal ideal in R . Then MRJJ= () x R for some ggg g x ∈HM(). Hence,RxRMRxMRJJJ==[( ) : ] [( ): ] J since M is gr-finitely gg g g generated. ThereforeRMRJJ=θ () and they by local property θ ()MR= . Conversely, suppose θ g ()MR= . Then there exist, g mm12,,,⋅⋅⋅ mk ∈ HN()such that RMmMmM==θ ()[():][():]12 + +⋅⋅⋅+ [():] mMk . Thus g MMMMmMMmMMmMmmmM=θ ( ) = [(12 ): ] + [( ): ] +⋅⋅⋅+ [(kn ): ] ⊆ ( 12 ) + ( ) +⋅⋅⋅+ ( ) ⊂ soMmmm=⋅⋅⋅()12,, k is gr-finitely generated. Let J be a gr-maximal ideal of R . Since θ g ()MR= , there is x ∈HM(), with [(x ):MJ ]Ш . Therefore, there exists rRJ∈− with rM⊆ () x and then ggggg gg rMRJJJJJ=< r > R ⋅ MR = MR ⊆ () x R . Hence MRJJ= () x R , for any gr-maximal ideal J of R and so M is locally gr-principal .

References

[1] Ameri, R., On The Prime Submodules of Multiplication Modules, Inter. J. of Mathematics and Mathematical Sciences, 27, 1715-1724 (2003).

[2] Anderson, D.D: some remark on multiplication ideal. Math Japonica 25, 463- 469,4(1980).

[3] El-Bast, Z. and Smith, P.F., Multiplication Modules, Communications in Algebra, 16(4), 755-779 (1988).

24 A. Khaksari and F. Rasti Jahromi

[4] Escoriza, J. and Torrecillas, B., Multiplication Graded Rings, Algebra and Number Theory, 127-136, Dekker, Lecture Notes in Pure and Appl. Math., 208, New York, 2000.

[5] Escoriza, J. and Torrecillas, B., Multiplication Objects in Commutative Grothendieck Categories, Communication in Algebra, 26(6), 1867-1883 (1998).

[6] Nastasescu, C. and Van oystaeyen, F: Graded Ring theory. Mathematical library , North Holand, Amesterdam, 28 (1982).

[7] Refai, M. Al Zoubi, K : On Graded Primary Ideals. Turkish Journal of Mathematics, 28, 217-229 (2004) .

Received: November, 2012