Different Types of G-Prime Ideals Associated to a Graded Module and Graded Primary Decomposition in a Graded Prüfer Domain

Abstract

In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it.

Graded Prime Ideals Attached to a Group Graded Module

“…[7] Let M be a G-graded A-module and N be a graded submodule of M . Then (1) The graded radical of N is denoted by Gr(N ) and defined as the ideal of A generated by the set {a ∈ h(A) :…”

“…[1, Definition 3.1] Let A be a G-graded integral domain. Then (1) A is said to be a G-graded valuation domain if for any x, y ∈ h(A), either x divides y or y divides x. (2) A is said to be a G-graded Prüfer domain if its graded localization A G p is a G-graded valuation domain for each G-prime ideal P of A.…”

“…We study the properties of G-graded non-Noetherian modules more closely by using G-attached and weak G-attached prime ideals. Recent works in this direction (i.e., concepts of G-associated prime ideals and graded primary decomposition) can be found in [1], [2], [7] and [11].…”