RINGS IN WHICH EVERY S-FINITE IDEAL IS S-FINITELY PRESENTED

RINGS IN WHICH EVERY S-FINITE IDEAL IS S-FINITELY PRESENTED

In this paper, we introduce and investigate rings in which every S-finite ideal is S-finitely presented, called strongly S-coherent rings. Any strongly S-coherent ring is S-coherent. It is clear that a ring is Noetherian if and only if it is S-Noetherian and strongly S-coherent. We study the transfer of this notion to various contexts of commutative ring extensions, such as direct products, trivial ring extensions, and pullbacks. Our results generate new families of examples of S-coherent rings that are not strongly S-coherent, and we show that the S-Noetherian and strongly S-coherent properties are not comparable.