On graded P-compactly packed 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 introduce the concept of graded P-compactly packed modules and we give a number of results concerning such graded modules. In fact, our objective is to investigate graded P-compactly packed modules and examine in particular when graded R-modules are P-compactly packed. Finally, we introduce the concept of graded finitely P-compactly packed modules and give a number of its properties.


Introduction and Preliminaries
Graded primary ideals in a commutative graded ring have been introduced and studied by M. Refai and K. Al-Zoubi in [8]. Graded primary submodules of graded modules over graded commutative rings have been studied in [3,4]. Graded primary radical of a graded submodule over graded commutative rings have been introduced and studied by K. Al-Zoubi in [1]. Also, the concept of graded compactly packed modules was introduced by F. Farzalipour and P. Ghiasvand in [4]. Here, we generalize this concept to the concept of graded P -compactly packed modules and give a number of its properties. We also introduce the concept of graded finitely P -compactly packed modules and give some results about it.
Before we state some results, let us introduce some notations and terminologies. 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. We denote this by .R; G/ (see [6].) 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 P g2G x g , where x g is the component of x in R g . Moreover, h.R/ D S g2G R g . Let I be an ideal of R. Then I is called a graded ideal of .R; G/ if I D L g2G .I T R g /. Thus, if x 2 I , then x D P g2G x g with x g 2 I . An ideal of a G-graded ring need not be G-graded (see [6].) 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 D L In this case, N g is called the g-component of N (see [6].) Let R be a G-graded ring and M a graded R-module. The graded radical of a graded ideal I , denoted by Gr.I /; is the set of all x D P g2G x g 2 R such that for each g 2 G there exists n g > 0 with x n g g 2 I: Note that, if r is a homogeneous element, then r 2 Gr.I / if and only if r n 2 I for some n 2 N. A proper graded ideal P of R is said to be graded primary ideal if whenever r; s 2 h.R/ with rs 2 P , then either r 2 P or s 2 Gr.P / (see [8].) A proper graded submodule N of a graded R-module M is said to be graded prime submodule if whenever r 2 h.R/ and m 2 h.M / with rm 2 N , then either r 2 .N W R M / D fr 2 R W rM Â N g or m 2 N . A proper graded submodule N of a graded R-module M is said to be graded primary submodule if whenever r 2 h.R/ and m 2 h.M / with rm 2 N , then either m 2 N or r 2 Gr..N W R M // (see [7].) The graded primary and primary submodules are different concepts (see [8,Example 1.6].) The graded radical of a graded submodule N of a graded R-module M; denoted by Gr M .N /; is defined to be the intersection of all graded prime submodules of M containing N . If N is not contained in any graded prime submodule of M , then Gr M .N / D M (see [7].) A graded R-module M is called graded finitely generated if there exist x g 1 ; x g 2 ; :::; x g n 2 h.M / such that M D Rx g 1 C C Rx g n : A graded R-module M is called graded cyclic if M D Rm g where m g 2 h.M /:

Graded P-compactly packed modules
In this section, we define the graded P -compactly packed modules and give a number of its properties. We also find the necessary and sufficient conditions for any graded R-module M to be graded P -compactly packed.  Proof. (i))(ii) Assume (i) holds and let N be a proper graded submodule of M . By [1, Theorem 2.4], P -Gr M .Rn g / Â P -Gr M .N / for each n g 2 N \ h.M /: Now, suppose that P -Gr M .N / ª P -Gr M .R n g / for each n g 2 N \ h.M /: Then for each n g 2 N \ h.M / there exists a graded primary submodule P n g for which Rn g Â P n g and N ª P n g : But N D [ n g 2N R n g Â [ n g 2N P n g ; that is M is not P -compactly packed, a contradiction.
(ii))(iii) Assume (ii) holds. Let N be a proper graded submodule of M and let fP˛g˛2 be a family of graded Proof. Let N be a proper graded submodule of M and let fP˛g˛2 be a family of graded primary submodules of M such that N Â [˛2 P˛: Since N is a graded cyclic, N D R n g for some n g 2 N \ h.M /: Since n g 2 N Â [˛2 P˛, n g 2 Pˇfor someˇ2 it follows that N D R n g Â Pˇ: Therefore M is graded P -compactly packed.
A graded R-module M is said to be with graded primary decomposition if each of its proper graded submodules is an intersection, possibly infinite, of graded primary submodules of M . Recall that a proper graded submodule N of a graded R-module M is said to be graded maximal submodule if there is no graded submodule K of M such that N ¦ K ¦ M (see [2].) Theorem 2.7. Let R be a G-graded ring and M a graded R-module. If M is graded P -compactly packed which has at least one graded maximal submodule, then M satisfies the ascending chain condition on graded primary radical submodules.
Proof. Let P 1 Â P 2 Â P 3 Â be an ascending chain of graded primary radical submodules of M . If P k D M for some k, then the result follows immediately, so assume that none of P k 's is M and let P D [ 1 iD1 P i : We claim that P is a proper graded submodule of M . Assume on contrary that P D M and let L be a graded maximal submodule of M . Then L Â [ 1 iD1 P i : Since M is graded P -compactly packed, by Theorem 2.3 L Â P k for some k. Hence L D P k and so P k is graded maximal. Hence P k D P i for all i k it follows that P k D [ 1 iD1 P i D M; which is impossible. Thus P is a proper graded submodule of M . Since M is graded P -compactly packed, by Theorem 2.3 P Â P s for some s and hence P s D P i for all i s: Therefore the ascending chain condition is satisfied on graded primary radical submodules.
By [2, Lemma 2.7], every graded finitely generated module over graded ring has a graded proper maximal submodule. Then we have the following Corollary.
Corollary 2.8. Let R be a G-graded ring and M a graded finitely generated R-module. If M is graded P -compactly packed, then M satisfies the ascending chain condition on graded primary radical submodules. Lemma 2.9. Let R be a G-graded ring and M a graded R-module. If M satisfies the ascending chain condition on graded primary radical submodules, then every graded primary radical submodule is the graded primary radical of a graded finitely generated submodule.
Proof. Assume that there exists a graded primary radical P which is not graded primary radical of a graded finitely generated submodule. Let n 1 2 P \ h.M / and let P 1 D P -Gr M .R n 1 /: Then P 1¨P : Hence there exists n 2 2 .P \ h.M // P 1 : Let P 2 D P -Gr M .R n 1 R n 2 /: Then P 1¨P2¨P and hence there exists n 3 2 .P \ h.M // P 2 etc. This gives an ascending chain of graded primary radical submodules P 1¨P2¨P3¨ which is a contradiction.
Theorem 2.10. Let R be a G-graded ring and M a graded R-module such that every graded finitely generated submodule of M is graded cyclic. If M satisfies the ascending chain condition on graded primary radical submodules, then M is a graded P -compactly packed.
Proof. Let N be a proper graded submodule of M . By Lemma 2.9, there exists a graded finitely generated submodule P of M such that P -Gr M .N / D P -Gr M .P /: By our assumption we conclude that P is a graded cyclic, it follows that there exists n g 2 N \ h.M / such that P D R n g : By Theorem 2.3, M is a graded P -compactly packed.
Let M and M 0 be two graded R-modules. A homomorphism of graded R-modules ' W M ! M 0 is a homomorphism of R-modules verifying '.M g / Â M 0 g for every g 2 G.
Lemma 2.11. Let R be a G-graded ring and M; M 0 be two graded R-modules and ' W M ! M 0 be an epimorphism of graded modules. If M is a graded P -compactly packed, then so is M 0 : Proof. Assume that M is a graded P -compactly packed. Let N 0 be a proper graded submodule of M 0 and let fP 0 g˛2 be a family of graded primary submodules of M 0 such that N 0 Â [˛2 P 0 . Since ' is an epimorphism of graded modules, Lemma 2.14], ' 1 .P 0 / is a graded primary submodule of M for each˛2 : Since M is a graded P -compactly packed, there existsˇ2 such that ' 1 .N 0 / Â ' 1 .P 0 /: Thus N 0 Â P 0 for someˇ2 : Therefore M 0 is a graded P -compactly packed. Since M 0 is a graded P -compactly packed, '.N / Â '.Pˇ/ for someˇ2 . Now, we show that N Â Pˇ: Let n D P g2G n g 2 N: for g 2 G; n g 2 N and so '.n g / 2 '.N / Â '.Pˇ/: Hence there exists t 2 Pˇ\ h.M / such that '.n g / D '.t /: Hence n g t 2 Ker.'/ Â Pˇ; it follows that n g 2 Pˇ: So N Â Pˇ: Therefore M is a graded P -compactly packed.
Let R be a G-graded ring and M a graded R-module and S Â h.R/ a multiplicatively closed subset of R. A non empty subset S of h.M / is said to be graded S -closed if se 2 S for every s 2 S and e 2 S (see [

Graded finitely P-compactly packed modules
In this section, we define the graded finitely P -compactly packed modules and give a number of its properties. Also, we find the conditions that make graded finitely P -compactly packed modules graded P -compactly packed. It is clear that if M is graded P -compactly packed, then M is graded finitely P -compactly packed.
Theorem 3.2. Let R be a G-graded ring and M a graded R-module in which every finite family of graded primary submodules of M is totally ordered by inclusion. If M is graded finitely P -compactly packed, then M is graded P -compactly packed.
Proof. Let N be a proper graded submodule of M and let fP˛g˛2 be a family of graded primary submodules of M such that N Â [˛2 P˛. Since M is graded finitely P -compactly packed, there exist˛1;˛2; : : : ;˛n 2 such that N Â [ n i D1 P˛i : Since fP˛i g n iD1 is totally ordered by inclusion, there existsˇ2 f˛1;˛2; : : : ;˛ng such that [ n iD1 P˛i D Pˇ: Thus M is graded P -compactly packed.
Let N 1 ; N 2 ; : : : N n be graded submodules of a graded R-module M . We call a covering N Â N 1 [ N 2 [ [ N n efficient if N is not contained in the union of any n 1 of the graded submodules N 1 ; N 2 ; : : : N n : Any covering of a union of graded submodules can be reduced to an efficient one, called an efficient reduction, by deleting any unnecessary terms, (see [3].) Theorem 3.3. Let R be a G-graded ring and M a graded multiplication R-module such that Gr M .N / D N for all graded submodules N of M . If M is graded finitely P -compactly packed, then M is graded P -compactly packed.
Proof. Let N be a proper graded submodule of M and let fP˛g˛2 be a family of graded primary submodules of M such that N Â [˛2 P˛. Since M is graded finitely P -compactly packed, there exist˛1;˛2; : : : ;˛n 2 such that N Â [ n i D1 P˛i : we may assume that the covering is efficient. We show that Gr..P j W R M // ª Gr..P k W R M // whenever j ¤ k: Assume on contrary that Gr..P j W R M // Â Gr..P k W R M // for some j ¤ k: By [7, Theorem 9], P j D Gr M .P j / D Gr..P j W R M //M Â Gr..P k W R M //M D Gr M .P k / D P k ; a contradiction. Thus Gr..P j W R M // ª Gr..P k W R M // whenever j ¤ k: By [3, Theorem 2.6], N Â Pˇfor someˇ: Therefore M is graded P -compactly packed.
Theorem 3.4. Let R be a G-graded ring and M a graded R-module. If M is graded finitely P -compactly packed which has at least one graded maximal submodule, then M satisfies the ascending chain condition on graded primary submodules.
Proof. Let P 1 Â P 2 Â P 3 Â be an ascending chain of graded primary submodules of M and let P D [ 1 i D1 P i : We claim that P is a proper graded submodule of M . Assume on contrary that P D M and let L be a graded maximal submodule of M . Then L Â [ 1 iD1 P i : Since M is graded finitely P -compactly packed, there exist m 1; m 2 ; : : : ; m k such that L Â [ k j D1 P m j D P m where m D maxfm 1; m 2 ; : : : ; m k g: Since L is graded maximal, L D P m : Hence P m is graded maximal, it follows that P i D P m for all i m: Thus P m D [ 1 i D1 P i D M which is impossible. Thus P is a graded proper submodule of M: Since M is graded finitely P -compactly packed, there exist t 1; t 2 ; : : : ; t n such that P Â [ n j D1 P t j D P t where t D maxft 1; t 2 ; : : : ; t n g: Hence P i Â P t for all i , thus P i D P t for all i t: Then the ascending chain condition is satisfied on graded primary submodules.