On nonnil S-SFT rings

Abstract

Let [Formula: see text] be a commutative ring with nonzero identity element and S a multiplicative subset of [Formula: see text]. In this paper, we introduce and investigate the notion of nonnil S-SFT rings. The ring [Formula: see text] is said to be nonnil S-SFT, if for each nonnil-ideal [Formula: see text] of [Formula: see text], there exist [Formula: see text], [Formula: see text] and a finitely generated ideal [Formula: see text] such that [Formula: see text] for all [Formula: see text], in that case, the ideal [Formula: see text] is called S-SFT. It is shown that the ring [Formula: see text] is nonnil S-SFT ring if and only if each nonnil-prime ideal (disjoint with S) is S-SFT. The transfert of the nonnil S-SFT concept to flat overrings and trivial extension is investigated. We give several characterizations of nonnil S-SFT rings. Also, we give a characterization of the nonnil-SFT variant in terms of the nonnil S-SFT variant.