TAILIEUCHUNG - Augmented Noetherian graded modules

This paper we study the augmented graded Noetherian modules and try to give some relationships between the Noetherian modules in the category R − Agr and the Noetherian modules in the category Re − gr. | Turk J Math 23 (1999) , 355 – 360. ¨ ITAK ˙ c TUB AUGMENTED NOETHERIAN GRADED MODULES Masshoor Refai Abstract Let G be a multiplicative group with identity e, and let R be an associative Ggraded ring with unity 1. In this paper we study the augmented graded Noetherian modules and try to give some relationships between the Noetherian modules in the category R − Agr and the Noetherian modules in the category Re − gr. Keywords and phrases: Graded Noetherian modules, graded rings, Augmented graded modules. Introduction Let G be a multiplicative group with identity e, and let R be an associative G-graded ring with unity 1. Let Re be the identity component of R, and let Re −gr be the category of all graded Re -modules and their graded Re -maps. In [5,6], we defined the concepts of augmented graded rings and augmented graded modules and we studied some of their properties. In this paper we use these concepts to study the augmented graded Noetherian modules. Some of the material in this paper are related to the work done by C. Nastasescu and F. Van Oystaeyen in [1, 2, 3, 4]. In Section 1, we give some definitions of graded rings and graded modules which are necessary in this paper. In Section 2, we discuss some facts concerning the augmented graded Noetherian modules, and we give some relationships between the Noetherian modules in the category R − Agr and the Noetherian modules in the category Re − gr. 355 REFAI 1. Preliminaries In this section we give some definitions of graded rings and graded modules which are necessary in this paper. For more details one can look in [5, 6]. Let G be a group with identity e, and R be a G-graded ring. We consider R − gr to be the category of all left graded R-modules and their graded R-maps. It is well known that R-gr is a Grothendieck category [3]. Also, we consider supp(R, G) = {g ∈ G : Rg 6= 0}. ⊕ Rg is said to be an augmented G-graded g∈G ring if it satisfies the following conditions : Definition . A G-graded ring R = ⊕

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