Rings in which every regular finitely generated ideal is $S$-finitely presented

Abstract

In this work, we introduce and explore a class of rings where every regular finitely generated ideal is $S$-finitely presented, called a regular $S$-coherent ring. This concept represents a weaker version of the $S$-coherent ring property. It is shown that any $S$-coherent ring is inherently a regular $S$-coherent ring, and in the case of domains, the two properties are equivalent. We also investigate how this notion extends to different settings of commutative ring extensions, including direct products, trivial ring extensions, and the amalgamated duplication of a ring along an ideal. The obtained results yield new examples of regular $S$-coherent rings that are not $S$-coherent.