Some properties of graded comultiplication modules

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 will obtain some results concerning the graded comultiplication modules over a commutative graded ring.


Introduction and preliminaries
Graded multiplication modules (gr-multiplication modules) over commutative graded ring have been studied by many authors extensively (see [1][2][3][4][5][6][7]). As a dual concept of gr-multiplication modules, graded comultiplication modules (gr-comultiplication modules) were introduced and studied by Ansari-Toroghy and Farshadifar [8]. A graded R-module M is said to be graded multiplication module (gr-multiplication module) if for every graded submodule N of M there exists a graded ideal I of R such that N D IM (see [3]). A graded R-module M is said to be graded comultiplication module (gr-comultiplication module) if for every graded submodule N of M there exists a graded ideal I of R such that N D .0 W M I /; where .0 W M I / D fm 2 M W mI D 0g (see [8]). Also it was shown that M is a gr-comultiplication module if and only if for each graded submodule N of M , N D .0 W M Ann R .N //. Here we will study the class of graded comultiplication modules and obtain some further results which are dual to classical results on graded multiplication modules (see Section 2).
First, we recall some basic properties of graded rings and modules which will be used in the sequel. We refer to [9] and [10] for these basic properties and more information on graded rings and modules. Let G be a group with identity e and R be a commutative ring with identity 1 R . Then R is a G-graded ring if there exist additive subgroups R g of R such that R D L g2G R g and R g R h Â R gh for all g; h 2 G. The elements of R g are called to be homogeneous of degree g where the R g 's are additive subgroups of R indexed by the elements g 2 G. If x 2 R, then x can be written uniquely as An ideal of a G-graded ring need not be G-graded.
Let R be a G-graded ring and M an R-module. We say that M is a G-graded R-module (or graded R-module) if there exists a family of subgroups fM g g g2G of M such that M Dg  N g where N g D N \ M g for g 2 G: In this case, N g is called the g-component of N .
Let R be a G-graded ring and M a graded R-module. A proper graded ideal I of R is said to be graded maximal ideal (or gr-maximal ideal) of R if J is a graded ideal of R such that I Â J Â R, then I D J or J D R.
A non-zero (resp. a proper) graded ideal I of a G-graded ring R is said to be gr-large (resp. gr-small) if for every non-zero (resp. proper) graded ideal J of R , we have I \ J ¤ 0 (resp. I C J ¤ R).
A graded submodule N of a graded R-module M is said to be graded minimal (gr-minimal) if it is minimal in the lattice of graded submodules of M: A non-zero (resp. a proper) graded submodule N of a graded R-module M is said to be gr-large (resp. grsmall) if for every non-zero (resp. proper) graded submodule L of M , we have N \ L ¤ 0 (resp. L C N ¤ M ).
A graded R-module M is said to be gr-uniform (resp. gr-hollow) if each of its proper graded submodules is gr-large (resp. gr-small).
A graded R-module M is said to be gr-simple if .0/ and M are its only graded submodules. A graded R-module M is said to be gr-faithful if aM D 0 implies a D 0 for a 2 h.R/.
A graded R-module M is said to be graded finitely generated if there exist x g1 ; x g2 ; :::; x gn 2 h.M / such that M D Rx g1 C CRx gn .
A graded R-module M is said to be gr-Artinian if satisfies the descending chain condition for graded submodules.
A proper graded ideal P of a G-graded ring R is said to be graded prime ideal (gr-prime ideal) if whenever r; s 2 h.R/ with rs 2 P , then either r 2 P or s 2 P (see [11]).
A non-zero graded submodule N of a graded R-module M is said to be a graded second (gr-second) if for each homogeneous element a of R, the endomorphism of M given by multiplication by a is either surjective or zero (see [8]).

Results
The following lemma is known (see [12] and [6]), but we write it here for the sake of references. Recall that a non-zero graded R-module M over a G-graded ring R is said to be gr-prime if Ann R .M / D Ann R .K/ for every non-zero graded submodule K of M (see [10]).
Theorem 2.2. Let R be a G-graded ring and M a graded R-module. If M is a gr-comultiplication gr-prime R-module, then M is a gr-simple module.
Proof. Let K be a non-zero graded submodule of M . Since M is gr-prime, we have Ann R .K/ D Ann R .M /. By [8, Lemma 3.5], we have K D M: Therefore M is a gr-simple module.
Recall that a graded R-module M is called finitely gr-cogenerated if for every non-empty family of graded submodules K r .r 2 L/ of M such that \ r2L K r D 0, there exists a finite subset F Â L verifying \ r2F K r D 0 (see [5]). Theorem 2.3. Let R be a G-graded ring and M a graded R-module. If M is finitely gr-cogenerated grcomultiplication R-module such that for each graded ideal J of R and for each collection fK˛g˛2 ƒ of graded submodules of M , we have .\˛2 ƒ K˛/J D \˛2 ƒ .K˛J /, then M is gr-Aritinian module.
This completes the proof because the reverse inclusion is clear.
Recall that a G-graded ring R is said to be a gr-comultiplication ring if it is a gr-comultiplication R-module (see [8]). (ii) Let N be a gr-minimal submodule of M and let K 1 , K 2 be two graded submodules of M such that Recall that a proper graded ideal I of a G-graded ring R is said to gr-irreducible if whenever I D I 1 \ I 2 for graded ideals I 1 and I 2 of R, then I D I 1 or I D I 2 (see [11]). Theorem 2.9. Let R be a G-graded ring and M a gr-comultiplication R-module such that Ann R .M / is grirreducible ideal of R. Then M is gr-hollow module.
Proof. Let K 1 and K 2 be graded submodules of M with M D K 1 C K 2 . Then Ann R .M / D Ann R .