Rings in which every $S$-finite ideal is $S$-finitely presented
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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.
Abstract
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.