When every regular ideal is S-finite

Abstract

In this paper, we introduce a new class of ring called regular S-Noetherian ring, which is a weak version of S-Noetherian ring property. Any S-Noetherian ring is naturally a regular S-Noetherian ring, and in the domain context, these two forms coincide. We study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamated duplication of a ring along an ideal. Our results generate new families of examples of non-S-Noetherian regular S-Noetherian rings.