$$v_c$$-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.

Integral Domains Whose w-Integral Closures Are Krull Domains

Let D be an integral domain with quotient field K, [Formula: see text] be the integral closure of D in K, and D[w] be the w-integral closure of D in K; so [Formula: see text], and equality holds when D is Noetherian or dim(D) = 1. The Mori–Nagata theorem states that if D is Noetherian, then [Formula: see text] is a Krull domain; it has also been investigated when [Formula: see text] is a Dedekind domain. We study integral domains D such that D[w] is a Krull domain. We also provide an example of an integral domain D such that [Formula: see text], t-dim(D) = 1, [Formula: see text] is a Prüfer v-multiplication domain with t-dim([Formula: see text]) = 2, and D[w] is a UFD.

Power Series Rings Over Prüfer v-multiplication Domains. II

Let D be an integral domain, X1(D) be the set of height-one prime ideals of D, {Xβ} and {Xα} be two disjoint nonempty sets of indeterminates over D, D[{Xβ}] be the polynomial ring over D, and D[{Xβ}][[{Xα}]]1 be the first type power series ring over D[{Xβ}]. Assume that D is a Prüfer v-multiplication domain (PvMD) in which each proper integral t-ideal has only finitely many minimal prime ideals (e.g., t-SFT PvMDs, valuation domains, rings of Krull type). Among other things, we show that if X1(D) = Ø or DP is a DVR for all P ∊ X1(D), then D[{Xβ}][[{Xα}]]1D−{0} is a Krull domain. We also prove that if D is a t-SFT PvMD, then the complete integral closure of D is a Krull domain and ht(M[{Xβ}][[{Xα}]]1) = 1 for every height-one maximal t-ideal M of D.

Divisor class groups and graded canonical modules of multisection rings

We describe the divisor class group and the graded canonical module of the multisection ringT(X;D1,…,Ds) for a normal projective varietyXand Weil divisorsD1,…,DsonXunder a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

Divisor class groups and graded canonical modules of multisection rings

We describe the divisor class group and the graded canonical module of the multisection ring T (X;D1,…, Ds) for a normal projective variety X and Weil divisors D1,…, Ds on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

Krull Modules

In this article, we generalize the concepts of several classes of domains (which are related to Krull domains) to torsion-free modules, and show that for a faithful multiplication module M over an integral domain R, M is a Krull module if and only if R is a Krull domain. Then we characterize Krull, Dedekind, and factorial modules.

THE RINGS $D((\mathcal{X}))_i$ AND $D\{\{\mathcal{X}\}\}_i$

Let D be an integral domain, [Formula: see text] be an infinite set of indeterminates over D, and [Formula: see text] be the i th type of power series ring over D for i = 1, 2, 3. For [Formula: see text], let c(f) denote the ideal of D generated by the coefficients of f. For a star operation * on D, put [Formula: see text], where *f is the star operation of finite type on D induced by *. Let [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We show that [Formula: see text], and that D is a Noetherian domain if and only if [Formula: see text] is a Noetherian domain. We also show that D is a Krull domain if and only if [Formula: see text] is a Dedekind domain, if and only if [Formula: see text] is a Prüfer domain, and that D is a Dedekind domain if and only if [Formula: see text] is a Dedekind domain, if and only if [Formula: see text] is a Prüfer domain.

PRESENTATIONS AND MODULE BASES OF INTEGER-VALUED POLYNOMIAL RINGS

Let D be an integral domain with quotient field K. For any set X, the ring Int (DX) of integer-valued polynomials onDX is the set of all polynomials f ∈ K[X] such that f(DX) ⊆ D. Using the t-closure operation on fractional ideals, we find for any set X a D-algebra presentation of Int (DX) by generators and relations for a large class of domains D, including any unique factorization domain D, and more generally any Krull domain D such that Int (D) has a regular basis, that is, a D-module basis consisting of exactly one polynomial of each degree. As a corollary we find for all such domains D an intrinsic characterization of the D-algebras that are isomorphic to a quotient of Int (DX) for some set X. We also generalize the well-known result that a Krull domain D has a regular basis if and only if the Pólya–Ostrowski group of D (that is, the subgroup of the class group of D generated by the images of the factorial ideals of D) is trivial, if and only if the product of the height one prime ideals of finite norm q is principal for every q.