When Are Graded Rings Graded S-Noetherian Rings

Abstract

Let Γ be a commutative monoid, R=⨁α∈ΓRα a Γ-graded ring and S a multiplicative subset of R0. We define R to be a graded S-Noetherian ring if every homogeneous ideal of R is S-finite. In this paper, we characterize when the ring R is a graded S-Noetherian ring. As a special case, we also determine when the semigroup ring is a graded S-Noetherian ring. Finally, we give an example of a graded S-Noetherian ring which is not an S-Noetherian ring.

Graded S-Noetherian Modules

“…More precisely, they proved that a G-graded ring A is G-graded Noetherian if and only if A is Noethrian, provided G is finitely generated. Inspired by it, Kim and Lim [10] introduced the notion of G-graded S-Noetherian ring and extended previous result to this class. A G-graded ring A = g∈G A g is called G-graded S-Noetherian, where S is a given m.c.s.…”

“…Thereafter S-Noetherian rings and modules were continuously studied by many authors (see [4], [6], [10], [11], [12], [13], [14], [15] and [16], for example). This notion has motivated many researchers to study S-version of known structures in ring and module theory (see [4], [6], [21] and [22], for example).…”