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On graded secondary modules

Let G be a group with identity e, and let R be a G-graded commutative ring. Here we study the graded primary submodules of a G-graded R-module and examine when graded submodules of a graded representable module are graded representable. A number of results concerningof these class of submodules are given. | Turk J Math 31 (2007) , 371 – 378. ¨ ITAK ˙ c TUB On Graded Secondary Modules S. Ebrahimi Atani and F. Farzalipour Abstract Let G be a group with identity e, and let R be a G-graded commutative ring. Here we study the graded primary submodules of a G-graded R-module and examine when graded submodules of a graded representable module are graded representable. A number of results concerning of these class of submodules are given. Key Words: Graded secondary modules, Graded primary submodules. 1. Introduction Secondary modules have been studied extensively by many authors (see [3], [6] and [2], for example). Here we study graded representable modules and the graded primary submodules of a graded module over a G-graded commutative ring. Various properties of such modules are considered. For example, we show that every graded primary submodule of a graded representable module over a G-graded ring is graded representable. Before we state some results let us introduce some notation and terminology. Let G be an arbitrary group with identity e. A commutative ring R with non-zero identity is G-graded if it has a direct sum decomposition (as an additive group) R = ⊕g∈G Rg such that 1 ∈ Re ; and for all g, h ∈ G, Rg Rh ⊆ Rgh . If R is G-graded, then an R-module M is said to be G-graded if it has a direct sum decomposition M = ⊕g∈G Mg such that for all g, h ∈ G, Rg Mh ⊆ Mgh . An element of some Rg or Mg is said to be homogeneous element. A submodule of N ⊆ M , where M is G-graded, is called G-graded if N = ⊕g∈G (N ∩ Mg ) or if, equivalently, N is generated by homogeneous elements. Moreover, M/N becomes 2000 AMS Mathematics Subject Classification: 13A02, 16W50 371 ATANI, FARZALIPOUR a G-graded module with g-component (M/N )g = (Mg + N )/N for g ∈ G. Clearly, 0 is a graded submodule of M . Also, we write h(R) = ∪g∈G Rg and h(M ) = ∪g∈G Mg . A graded ideal I of R is said to be a graded prime ideal if I = R; and whenever ab ∈ I, we have a ∈ I or b ∈ I, where a, b ∈ h(R).