GRADED w-NOETHERIAN MODULES OVER GRADED RINGS

In this paper, we study the basic theory of the category of graded w-Noetherian modules over a graded ring R. Some elementary concepts, such as w-envelope of graded modules, graded w-Noetherian rings and so on, are introduced. It is shown that: (1) A graded domain R is graded w-Noetherian if and only if R-m(g) is a graded Noetherian ring for any gr-maximal w-ideal m of R, and there are only finite numbers of gr-maximal w-ideals including a for any nonzero homogeneous element a. (2) Let R be a strongly graded ring. Then R is a graded w-Noetherian ring if and only if R-e is a w-Noetherian ring. (3) Let R be a graded w-Noetherian domain and let a is an element of R be a homogeneous element. Suppose p is a minimal graded prime ideal of (a). Then the graded height of the graded prime ideal p is at most 1.