When Every S-Finite Ideal is Finitely Generated

In this paper, we introduce and define the concept of SFG-ring which is a ring for which every S-finite ideal is finitely generated. We investigate this property in various contexts of commutative ring extensions including direct product, trivial ring extensions, pullbacks and amalgamation rings. We provide examples to illustrate that this SFG-property is in fact a new and distinct class of rings. We also introduce the notion of nonnil SFG-ring and establish several results.