Almost graded multiplication and almost graded comultiplication modules

Abstract Let G be a group with identity e, R be a G-graded commutative ring with a nonzero unity 1 and M be a G-graded R-module. In this article, we introduce and study the concept of almost graded multiplication modules as a generalization of graded multiplication modules; a graded R-module M is said to be almost graded multiplication if whenever a ∈ h ( R ) a\in h(R) satisfies Ann R ( a M ) = Ann R ( M ) {\text{Ann}}_{R}(aM)={\text{Ann}}_{R}(M) , then ( 0 : M a ) = { 0 } (0{:}_{M}a)=\{0\} . Also, we introduce and study the concept of almost graded comultiplication modules as a generalization of graded comultiplication modules; a graded R-module M is said to be almost graded comultiplication if whenever a ∈ h ( R ) a\in h(R) satisfies Ann R ( a M ) = Ann R ( M ) {\text{Ann}}_{R}(aM)={\text{Ann}}_{R}(M) , then a M = M aM=M . We investigate several properties of these classes of graded modules.

Abstract: Let G be a group with identity e, R be a G-graded commutative ring with a nonzero unity 1 and M be a G-graded R-module. In this article, we introduce and study the concept of almost graded multiplication modules as a generalization of graded multiplication modules; a graded R-module M is said to be almost graded multiplication if whenever ∈ ( ) a h R satisfies ( ) = ( ) aM M Ann Ann

Introduction
Throughout this article, G will be a group with identity e and R a commutative ring with a nonzero unity 1. R is said to be G-graded if , where R g is an additive subgroup of R for all ∈ g G. The elements of R g are called homogeneous of degree g. If ∈ x R, then x can be written as ∑ ∈ x g G g , where x g is the component of x in R g . Also, we set ( ) = ⋃ ∈ h R R g G g . Moreover, it has been proved in [1] that R e is a subring of R and ∈ R 1 e . Let I be an ideal of a graded ring R. Then I is said to be graded ideal if ⊕ = ( ∩ ) ∈ I I R g G g , i.e., for ∈ x I, = ∑ ∈ x x g G g , where ∈ x I g for all ∈ g G. An ideal of a graded ring need not be graded; see the following example: is an ideal of R with + ∈ i I 1 . If I is G-graded, then ∈ I 1 , so = ( + ) a i 1 1 for some ∈ a R, i.e., = ( + )( + ) Let R be a G-graded ring and I be a graded ideal of R. Then R/I is G-graded by ( / ) = ( + )/ R I , where M g is an additive subgroup of M for all ∈ g G. The elements of M g are called homogeneous of degree g. It is clear that M g is an R e -submodule of M for all ∈ g G. Moreover, we set ( ) = ⋃ ∈ h M M g G g .
Let N be an R-submodule of a graded R-module M. Then N is said to be graded R-submodule if = A graded R-module M is said to be graded multiplication if for every graded R-submodule N of M, = N IM for some graded deal I of R. In this case, it is known that = ( ) I N M : R . Graded multiplication modules were first introduced and studied by Escoriza and Torrecillas in [3], and further results were obtained by several authors, see for example [4]. In [5], Atani introduced the concept of graded prime submodules; a proper graded R-submodule N of a graded R-module M is said to be graded prime if whenever ∈ ( ) r h R and ∈ ( ) . In [6], a graded R-module M is said to be graded weak multiplication if ( ) = M ϕ GSpec or for every graded prime R-submodule N of M, = N IM for some graded deal I of R.
Graded semiprime submodules have been introduced by Lee and Varmazyar in [7]. A proper graded R-submodule N of M is said to be graded semiprime if whenever I is a graded ideal of R and K is a graded R-submodule of M such that ⊆ I K N n for some positive integer n, then Graded semiprime submodules are also studied in [8]. The set of all graded semiprime R-submodules of M is denoted by . Motivated from the concepts of graded multiplication modules in [3] and graded weak multiplication modules in [6], a new class of graded R-modules has been introduced in [9], called graded semiprime multiplication modules. A graded R-module M is said to be graded semiprime multiplication if ( ) = ∅ M GSSpec or for every graded semiprime R-submodule N of M, = N IM for some graded ideal I of R. In [10], Atani introduced the concept of graded weakly prime submodules over graded commutative rings; where a graded proper R-submodule N of a graded R-module M is said to be graded weakly prime R-submodule of M if whenever ∈ ( ) r h R and ∈ ( ) m h M such that ≠ ∈ rm N 0 , then either ∈ m N or ∈ ( ) r N M : R . One can easily see that every graded prime submodule is graded weakly prime. However, the converse is not true in general; for example, { } 0 is graded weakly prime submodule by definition but { } 0 need not be graded prime submodule. In [11], several results on graded weakly prime submodules have been proved and investigated to introduce the concept of graded quasi multiplication modules; a graded R-module M is said to be graded quasi multiplication if for every graded weakly prime R-submodule N of M, . This concept has been first introduced and studied in [12], and then generalized into graded n-absorbing submodules in [13]. In [11], a parallel study given in [9] has been followed to investigate the new class of graded absorbing multiplication modules, by first providing many interesting results on graded 2-absorbing submodules. A graded R-module M is said to be graded absorbing multiplication if ( ) = M ϕ GABSpec or for every graded 2-absorbing R-submodule N of M, = N IM for some graded deal I of R.
So, most of all generalizations for graded multiplication modules were fixing on changing the graded R-submodule N from a general graded submodule to graded prime, graded semiprime, graded weakly prime or graded 2-absorbing submodule. In [14], an R-module M is said to be a quasi multiplication module if whenever In this article, we follow [14] to explore another technique to generalize the concept of graded multiplication modules. First, we need to introduce the following: Proof. Clearly, ( ) Similarly, one can prove the following: In this article, we introduce and study the concept of almost graded multiplication modules; a graded R-module M is said to be almost graded multiplication if whenever ∈ ( ) a h R satisfies . A generalization for graded comultiplication modules has been introduced and studied in [16]; a graded R-module M is said to be graded weak comultiplication if for every graded prime R-submodule N of M, = ( ) N I 0 : M for some graded ideal I of R. In [14], an R-module M is said to be a quasi comultiplication module if whenever

Graded comultiplication modules have been introduced by Toroghy and Farshadifar in
In this article, we follow [14] to introduce and study the concept of almost graded comultiplication modules; a graded R-module M is said to be almost graded comultiplication if whenever ∈ ( ) a h R satisfies

Almost graded multiplication modules
In this section, we introduce and study the concept of almost graded multiplication modules as a generalization of graded multiplication modules.
In [17], a graded R-module M is said to be graded Hopfian (resp. graded co-Hopfian) if every surjective (resp. injective) graded R-endomorphism of M is a graded R-isomorphism. . It follows that ∈ ( / ( )) a Z R R M Ann G , as needed. □ such that bM is an almost graded multiplication R-module, then M is an almost graded multiplication R-module.
, and then by assumption, we have that ( Remark 2.9. The converse of Proposition 2.8 is not true in general, because if it is true, then by Proposition 2.2, we have that every graded multiplication module is graded prime which is not true, as { } 0 is a graded multiplication module which is not a graded prime module. The next proposition shows that the converse of Proposition 2.8 will be true if R is integral domain and M is faithful. Proposition 2.10. Let M be an almost graded multiplication R-module. If R is an integral domain and M is faithful, then M is a graded prime R-module. defined by ( ) = f x ax is either surjective or zero. Graded second submodules have been wonderfully studied by Çeken and Alkan in [19]. The next proposition shows that the converse of Proposition 2.8 will be true if M is a graded second R-module.

Almost graded comultiplication modules
In this section, we introduce and study the concept of almost graded comultiplication modules as a generalization of graded comultiplication modules. such that ≠ aM M, which implies that M is not an almost graded comultiplication R-module.
The next example shows that not every almost graded comultiplication module is almost graded multiplication.   In [20], a proper -graded R-submodule N of M is said to be graded completely irreducible if whenever = ⋂ ∈ N N k k Δ where { } ∈ N k k Δ is a family of -graded R-submodules of M, then = N N k for some ∈ k Δ. In [21], the concept of graded completely irreducible submodules has been extended into G-graded case, for any group G. It has been proved that every graded R-submodule of M is an intersection of graded completely irreducible R-submodules of M. In many instances, we use the following basic fact without further discussion.
Remark 3.8. Let N and L be two graded R-submodules of M. To prove that ⊆ N L, it is enough to prove that if K is a graded completely irreducible R-submodule of M such that ⊆ L K , then ⊆ N K .