tailieunhanh - On graded prime and primary submodules

Let G be a multiplicative group. Let R be a G-graded commutative ring and M a G-graded R-module. Various properties of graded prime submodules and graded primary submodules of M are discussed. We have also discussed the graded radical of graded submodules of multiplication graded R-modules. | Turk J Math 35 (2011) , 159 – 167. ¨ ITAK ˙ c TUB doi: On graded prime and primary submodules ¨ K¨ ur¸sat Hakan Oral, Unsal Tekir and Ahmet G¨ oksel A˘garg¨ un Abstract Let G be a multiplicative group. Let R be a G -graded commutative ring and M a G -graded R -module. Various properties of graded prime submodules and graded primary submodules of M are discussed. We have also discussed the graded radical of graded submodules of multiplication graded R -modules. Key Words: Multiplication graded modules, graded prime submodules, graded primary submodules. 1. Introduction Let G be a multiplicative group with identity e and R be a commutative ring with identity. Then R is called a G -graded ring if there exist additive subgroups Rg of R indexed by the elements g ∈ G such that R = ⊕ Rg and Rg Rh ⊆ Rgh for all g, h ∈ G . We denote this by G (R) . The elements of Rg are called g∈G homogeneous of degree g and all the homogeneous elements are denoted by h (R), . h (R) = ∪ Rg . If g∈G ag , where ag is called the g -component of a in Rg . a ∈ R , then the element a can be written uniquely as g∈G In this case, Re is a subring of R and 1R ∈ Re . 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 {Mg }g∈G of M such that M = ⊕ Mg (as abelian groups) and g∈G Rg Mh ⊆ Mgh for all g, h ∈ G . Here, Rg Mh denotes the additive subgroup of M consisting of all finite sums of elements rg sh with rg ∈ Rg and sh ∈ Mh . Also, we write h (M ) = ∪ Mg and the elements of h (M ) are g∈G called homogeneous. If M = ⊕ Mg is a graded R -module, then for all g ∈ G the subgroup Mg of M is an g∈G Re -module. Let M = ⊕ Mg be a graded R -module and N a submodule of M . Then N is called a graded g∈G submodule of M if N = ⊕ Ng where Ng = N ∩ Mg for g ∈ G . In this case, Ng is called the g -component of g∈G N . Moreover, M/N becomes a G -graded R -module with g -component (M/N )g