$$v_c$$-Noetherian domains and Krull domains

Let D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

Two applications of Nagata rings and modules

Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

S-∗w-Principal Ideal Domains

Let D be an integral domain, ∗ a star-operation on D, and S a multiplicative subset of D. We define D to be an S-∗w-principal ideal domain if for each nonzero ideal I of D, there exist an element s ∈ S and a principal ideal (c) of D such that [Formula: see text]. In this paper, we study some properties of S-∗w-principal ideal domains. Among other things, we study the local property, the Nagata type theorem, and the Cohen type theorem for S-∗w-principal ideal domains.

Kronecker function rings of ring extensions

The classical Kronecker function ring construction associates to a domain [Formula: see text] a Bézout domain. Let [Formula: see text] be a subring of a ring [Formula: see text], and let ⋆ be a star operation on the extension [Formula: see text]. In their book [Manis Valuations and Prüfer Extensions II, Lectures Notes in Mathematics, Vol. 2103 (Springer, Cham, 2014)], Knebusch and Kaiser develop a more general construction of the Kronecker function ring of [Formula: see text] with respect to ⋆. We characterize in several ways, under relatively mild assumption on [Formula: see text], the Kronecker function ring as defined by Knebusch and Kaiser. In particular, we focus on the case where [Formula: see text] is a flat epimorphic extension or a Prüfer extension.

Overrings of Prüfer v-multiplication domains

Let [Formula: see text] be an integral domain, [Formula: see text] and [Formula: see text] the set of fractional ideals of [Formula: see text]. Let [Formula: see text] a finitely generated ideal with [Formula: see text]. For a torsion-free [Formula: see text]-module [Formula: see text], define [Formula: see text] for some [Formula: see text]. Call [Formula: see text] a [Formula: see text]-module if [Formula: see text]. On [Formula: see text], the function [Formula: see text] is a star-operation of finite character. An integral ideal [Formula: see text] maximal with respect to being a proper [Formula: see text]-ideal is a prime ideal called a maximal [Formula: see text]-ideal. A torsion-free [Formula: see text]-module [Formula: see text] is called [Formula: see text]-flat, if [Formula: see text] is a flat [Formula: see text]-module for each [Formula: see text], the set of maximal [Formula: see text]-ideals of [Formula: see text]. [Formula: see text] is called a Prüfer [Formula: see text]-multiplication domain (P[Formula: see text]MD), if [Formula: see text] is a valuation ring for each [Formula: see text]. We characterize [Formula: see text]-flat modules in a manner similar to the characterization of flat modules, study them when they are rings [Formula: see text] with [Formula: see text] and characterize P[Formula: see text]MDs using them and compare our work with similar work in the literature.