Graded Primary Submodules over Multiplication Modules

Let G be an abelian group with identity e, R be a G − graded commutative ring and M a graded R − module where all modules are unital. Various generalizations of graded prime ideals and graded submodules have been studied. For example, a proper graded ideal I is a graded weakly (resp; almost) prime ideal if 0 ≠ ab ∈ I (resp; ab ∈ I − I2) then a ∈ I or b ∈ I. Also a proper graded submodule N of M is graded primary submodule if rm ∈ N, then either m ∈ N or r∈(N:M). Throughout this work, we define that a proper graded submodule is a graded weakly (resp; almost) primary submodule if 0 ≠ rm ∈ N (resp; rm ∈ N − (N : M)N), then either m ∈ N or r∈(N:M). We give some properties and characterizations of graded weakly (resp; almost) primary submodules. We show that graded weakly primary submodules enjoy analogs of many of the properties of prime submodules and primary submodules. Mathematics Subject Classification: 13A02, 16W50.

. We give some properties and characterizations of graded weakly (resp; almost) primary submodules. We show that graded weakly primary submodules enjoy analogs of many of the properties of prime submodules and primary submodules.

INTRODUCTION
Several authors have extended the notion of prime ideals to modules [2], [3]. Almost prime ideals were introduced by S. M. Bhatwadekar and P. K. Sharma [6]. Graded almost prime ideals in a graded commutative ring with non-zero identity have been introduced and studied by A. Jabeer, M. Bataineh and H. Khashan [1,4]. Moreover, graded primary submodules in a graded commutative ring with non-zero identity have been studied by many authors such as S. Ebrahimi Atani [7] and K. H. Oral, V. Tekir and A. G. Agargun [5]. In this study, we introduce some properties and characterizations of graded weakly (almost) primary submodules.

2.GRADED WEAKLY PRIMARY SUBMODULES
Graded weakly primary ideals in a graded commutative ring with non-zero identity have been introduced and studied by S. E. Atani [8]. Graded primary submodules in a graded module over a graded commutative ring had been studied by many authors like K. H. Oral, U. Tekir and A. G. Agargun [5] and S. E. Atani [7]. In this section we study graded weakly (resp.,almost) primary submodules under multiplication and torsion free modules. Now we give the definition of a graded weakly primary and graded almost primary submodules over a graded commutative ring. for some positive integer k . Clearly a graded weakly (resp., almost) prime submodule is a graded weakly (resp., almost) primary submodule, but the converse need not to be true.
Next we give an example of a graded weakly primary submodule which is not graded weakly prime.
. We note that N is a graded weakly primary submodule, but not graded weakly prime since The following theorem characterizes the homogeneous components of a graded weakly primary submodules. Theorem 2.3 Let N be a graded submodule of M and G g ∈ . Then the following are equivalent.
(1) Ng is a weakly g -primary submodule of Mg .
Let P be an ideal of e R and K be a submodule of Mg such that By assuming ) and so a ) : . By previous case, we have 0 Mg Ng a n ∈ for some positive integer n or . Therefore, Ng is a weakly g -primary submodule of Mg for all G g ∈ . Theorem 2.5 Assume that N and K are graded submodules of M such that N K ⊆ with M N ≠ . Then (1) If N is a graded weakly primary submodule of M , then / is also a graded weakly primary submodule of / . (2) If K and / are graded weakly primary submodules, then N is also a graded weakly primary submodule of M .
proof (1) Let and K is a graded weakly primary submodule then either ) : is a graded weakly primary submodule we get either N m ∈ or ( ) for some positive integer k .
for some positive integer k , since N is a graded almost primary submodule in M . Then In the next theorems, we give a characterizations of graded weakly primary prime submodules in one kind of cancellation graded modules. We need the following definitions.
, since M is a cancellation module which is a contradiction. Since ) : ( M N is a graded almost primary ideal, we obtain ) : . Therefore, N is a graded almost primary submodule.

Theorem 2.12 Let M be a finitely generated faithful multiplication graded R -module and N be a proper graded submodule of M . Then N is a graded almost primary submodule of M if and only if whenever
which is a contradiction. By assuming either is a graded almost primary ideal and so N is a graded almost primary submodule of M .

Lemma 2.17 Let R be a graded domain and I a graded ideal of R . If I is a graded weakly primary ideal of R then I is a graded weakly prime ideal of R .
proof . Then there is some positive since R is a graded domain. As I a n ∉ , I graded weakly primary gives . Therefore I is a graded weakly prime ideal of R . proof Let )