On graded primary-like submodules of graded modules over graded commutative rings

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 primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to find extra properties of these graded submodules.


Introduction
Recently, H. F. Moghimi and F. Rashedi, in [18] studied primarylike submodules as a new generalization of primary ideals to modules. Also, the concept of primeful module was introduced and studied by C.P. Lu in [16].
The scope of this paper is devoted to the theory of graded modules over graded commutative rings. One use of rings and modules with gradings is in describing certain topics in algebraic geometry. Here, in particular, we are dealing with graded primary-like submodules.
The concept of graded primary ideal was introduced and studied by M. Refai and K. Al-Zoubi in [23].
In the literature, there are several different generalization of the notion of graded primary ideal to graded module. The concept of graded primary submodule was introduced by S.E. Atani and F. Farzalipour in [12] and studied in [1,4,14,22]. Also the the concept of graded prime submodule was introduced by S.E. Atani in [7] and studied in [2,3,5,6,9,10,17,22,25].
Here, we introduce the concept of graded primary-like submodule as a new generalization of a graded primary ideal on the one hand and a generalization of a graded prime submodule on other hand.
Our article is organized as follows.
In Section 2 we recall important notions which will be used throughout the paper. In Section 3 we will investigate graded submodules which satisfy the gr-primeful property. In Section 4 we introduce the concept of graded primary-like submodules and give a number of results concerning such modules. For example, we give a characterization of graded primary-like submodules. We also study the behavior of graded primary-like submodules under graded homomorphisms and under localization.

Preliminaries
Convention. Throughout this paper all rings are commutative with identity and all modules are unitary.
First, we recall some basic properties of graded rings and modules which will be used in the sequel. We refer to [15], [19], [20] and [21] 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 = g∈G R g and R g R h ⊆ R gh for all g, h ∈ 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 ∈ G. If x ∈ R, then x can be written uniquely as g∈G x g , where x g is the component of x in R g . Moreover, h(R) = g∈G R g . Let I be an ideal of R. Then I is called a graded ideal of (R, G) if I = g∈G (I R g ). Thus, if x ∈ I, then x = g∈G x g with x g ∈ I. 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 {M g } g∈G of M such that M = g∈G M g (as abelian groups) and R g M h ⊆ M gh for all g, h ∈ G. Here, R g M h denotes the additive subgroup of M consisting of all finite sums of elements r g s h with r g ∈ R g and s h ∈ M h . Also, we write h(M) = In this case, N g is called the g-component of N.
Let R be a G-graded ring and S ⊆ h(R) be a multiplicatively closed subset of R. Then the ring of fraction S −1 R is a graded ring which is called the graded ring of fractions. Indeed, Let R be a G-graded ring and M a graded R-module. A proper graded ideal I of R is said to be a graded maximal ideal of R if J is a graded ideal of R such that I ⊆ J ⊆ R, then I = J or J = R (see [24].) A proper graded ideal I of R is said to be a graded prime ideal if whenever rs ∈ I, we have r ∈ I or s ∈ I, where r, s ∈ h(R) (see [24].) The graded radical of I, denoted by Gr(I), is the set of all x = g∈G x g ∈ R such that for each g ∈ G there exists n g ∈ Z + with x ng ∈ I. Note that, if r is a homogeneous element, then r ∈Gr(I) if and only if r n ∈ I for some n ∈ N (see [24].) It is shown in [24, Proposition 2.5] that Gr(I) is the intersection of all graded prime ideals of R containing I. A proper graded ideal P of R is said to be a graded primary ideal if whenever r, s ∈ h(R) with rs ∈ P , then either r ∈ P or s ∈Gr(P ) (see [23].) A proper graded submodule N of M is said to be a graded prime submodule if whenever r ∈ h(R) and m ∈ h(M) with rm ∈ N, then either r ∈ (N : R M) = {r ∈ R : rM ⊆ N} or m ∈ N (see [7].) A proper graded submodule N of a graded R-module M is said to be a graded primary submodule if whenever r ∈ h(R) and m ∈ h(M) with rm ∈ N, then either m ∈ N or r ∈Gr((N : R M)) (see [12].) The graded radical of a graded submodule N of 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) = M (see [12].) A graded R-module M over G-graded ring R is said to be a 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 = IM. It is clear that M is gr-multiplication R-module if and only if N = (N : R M)M for every graded submodule N of M (see [13].)

Graded submodules which satisfy the gr-primeful property
The following Lemma is known (see [17, Lemma 1.2 and Lemma 2.7]), we write it her for the sake of references.   Let R be a G-graded ring and M, M ′ graded R-modules. Let ϕ : M → M ′ be an R-module homomorphism. Then ϕ is said to be a graded homomorphism if ϕ(M g ) ⊆ M ′ g for all g ∈ G (see [19].) Theorem 3.6. Let R be a G-graded ring and M, M ′ be two graded R-modules and N ′ a graded submodule of M ′ . Let ϕ : M → M ′ be a graded epimorphism. If N ′ satisfies the gr-primeful property, then so does ϕ −1 ( N ′ ).
Since N ′ satisfies the gr-primeful property, there exists a graded prime submodule P ′ of M ′ containing N ′ such that (P ′ : R M ′ ) = p. By [11, Lemma 5.2(ii)], we have that ϕ −1 ( P ′ ) is a graded prime submodule of M containing ϕ −1 ( N ′ ). It is easy to see (ϕ −1 (P ′ ) : R M) = p. Therefore ϕ −1 ( N ′ ) satisfies the gr-primeful property.  Proof. Let p be a graded prime ideal of R such that ( we have (N j : R M) ⊆ p for some j ∈ {1, 2, ..., n}. Since N j satisfies the gr-primeful property, there exists a graded prime submodule P of M containing N j with (P : R M) = p and so ∩ n i=1 N i ⊆ P. Thus ∩ n i=1 N i satisfies the gr-primeful property.  Proof. Suppose that rs ∈ (N : R M) and s / ∈ (N : R M) for some r, s ∈ h(R). We show that r ∈ (Gr(N) : R M). Let m = g∈G m g ∈ M.

Gragded primary-like submodules
Hence g∈G rsm g ∈ N. Since N is a graded primary-like submodule of M, rsm g ∈ N and s / ∈ (N : R M) for all g ∈ G, we conclude that rm g ∈Gr M (N) for all g ∈ G. Hence rm ∈Gr M (N). This shows that r ∈ (Gr(N) : M). Since N satisfies the gr-primeful property, by        Proof. Suppose that r g m h ∈ ϕ −1 ( N ′ ) and r g / ∈ (ϕ −1 ( N ′ ) : R M) for some r g ∈ h(R) and m h ∈ h(M). Hence ϕ(r g m h ) = r g ϕ(m h ) ∈ N ′ . Since r g / ∈ (ϕ −1 ( N ′ ) : R M), we get r g / ∈ ( N ′ : R M ′ ). Since N ′ is a graded primary-like submodule of M ′ , r g ϕ(m h ) ∈ N ′ and r g / By [11,Theorem 5.3 Proof. The proof is similar to the proof of [11,Theorem 5.3(ii)], so we omit it. Proof. Suppose that rm ′ ∈ ϕ( N) and r / ∈ (ϕ( N) : R M ′ ) for some r ∈ h(R) and m ′ ∈ h(M ′ ). It is easy to see r / ∈ (N : R M). Since ϕ is a graded epimorphism, there exists m ∈ h(M) such that ϕ(m) = m ′ and hence rϕ(m) = ϕ(rm) ∈ ϕ( N). Thus there exists n ∈ h(N) such that ϕ(rm) = ϕ( n) this implies rm − n ∈Kerϕ ⊆ N. This yields that rm ∈ N. Since N is a graded primary-like submodule of M, rm ∈ N and r / ∈ (N : R M)), we conclude that m ∈ G M (N). Hence ϕ(m) = m ′ ∈ ϕ(G M (N)). By Lemma 4.10, we have m ′ ∈ ϕ(G M (N)) = G M ′ (ϕ(N)). Therefore ϕ(N) is a graded primary-like submodule of M ′ . The following results study the behavior of graded primary-like submodules under localization.