Some characterizations of generalized coherent rings induced by a multiplicatively closed set

Abstract

In this paper, [Formula: see text] denotes a multiplicatively closed set of a commutative ring [Formula: see text]. Then [Formula: see text]-coherent rings are introduced. A ring [Formula: see text] is called [Formula: see text]-coherent if every finitely generated [Formula: see text]-ideal of [Formula: see text] is finitely presented. It is proved that if [Formula: see text] is a coherent domain with quotient field [Formula: see text], [Formula: see text], where [Formula: see text], and [Formula: see text], then [Formula: see text] is [Formula: see text]-coherent, but not coherent. And [Formula: see text]-flat modules are introduced to characterize [Formula: see text]-coherent rings, i.e., [Formula: see text] is [Formula: see text]-coherent if and only if every direct product of ([Formula: see text]-)flat modules is [Formula: see text]-flat; if and only if every finitely generated [Formula: see text]-module is finitely presented.