Graded Prime Ideals Attached to a Group Graded Module

Let G be a nitely generated abelian group and M be a G-graded A-module. In general, G-associated prime ideals to M may not exist. In this paper, we introduce the concept of G-attached prime ideals to M as a generalization of G-associated prime ideals which gives a connection between certain G-prime ideals and G-graded modules over a (not necessarily G-graded Noetherian) ring. We prove that the G- attached prime ideals exist for every nonzero G-graded module and this generalization is proper. We transfer many results of G-associated prime ideals to G-attached prime ideals and give some applications of it.