Primary decomposition over rings graded by finitely generated Abelian groups

Let A be a Noetherian ring which is graded by a finitely generated Abelian group G. In general, for G-graded modules there do not exist primary decompositions which are graded themselves. This is quite different from the case of gradings by torsion free group, for which graded primary decompositions always exists. In this paper we introduce G-primary decompositions as a natural analogue to primary decomposition for G-graded A-modules. We show the existence of G-primary decomposition and give a few characterizations analogous to Bourbakiʹs treatment for torsion free groups.