When Are Graded Rings Graded <i>S</i>-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.

Keywords

• <i>S</i>-Noetherian ring
• graded <i>S</i>-Noetherian ring
• <i>S</i>-finite algebra
• Cohen type theorem