Krull Domain Research Articles

V_c-Noetherian domains and Krull domains

Let D be an integrally closed domain, {V_{alpha }} be the set of t-linked valuation overrings of D, and v_c be the star operation on D defined by I^{v_c} = bigcap _{alpha } IV_{alpha } for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a v_c-Noetherian domain if and only if D is a Krull domain, if and only if 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 if and only if D is a Dedekind domain.

$\star$-super potent domains

For a finite-type star operation ⋆ on a domain R, we say that R is ⋆-super potent if each maximal ⋆-ideal of R contains a finitely generated ideal I such that (1) I is contained in no other maximal ⋆-ideal of R and (2) J is ⋆-invertible for every finitely generated ideal J⊇I. Examples of t-super potent domains include domains each of whose maximal t-ideals is t-invertible (e.g., Krull domains). We show that if the domain R is ⋆-super potent for some finite-type star operation ⋆, then R is t-super potent, we study t-super potency in polynomial rings and pullbacks, and we prove that a domain R is a generalized Krull domain if and only if it is t-super potent and has t-dimension one.

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.

On the generalized Krull property in power series rings

One open problem in commutative algebra and field arithmetic posed by Jarden is whether the power series ring R〚X〛 is a generalized Krull domain if R is a generalized Krull domain. Assuming R is a generalized Krull domain, Paran and Temkin proved that R〚X〛 is a generalized Krull domain if and only if R〚X〛 is a Krull domain. Hence, if R is a generalized Krull domain that is not a Krull domain, then R〚X〛 is never a generalized Krull domain. In this paper, we show that the assumption R is a generalized Krull domain in Paran and Temkin's result can be dropped. In other words, R〚X〛 is a generalized Krull domain if and only if R〚X〛 is a Krull domain and hence if and only if R is a Krull domain.

Kronecker function rings and power series rings

Let D be an integral domain with quotient field K , X be an indeterminate over D , D [ [ X ] ] be the power series ring over D , and c ( f ) be the ideal of D generated by the coefficients of f ∈ D [ [ X ] ] . We will say that a star operation ∗ on D is a c-star operation if (i) c ( f g ) ∗ = ( c ( f ) c ( g ) ) ∗ for all 0 ≠ f , g ∈ D [ [ X ] ] and (ii) ( A B ) ∗ ⊆ ( A C ) ∗ implies B ∗ ⊆ C ∗ for all nonzero fractional ideals A , B , C of D . Assume that D admits a c-star operation ∗ , and let Kr ( ( D , ∗ ) ) = { f ∕ g ∣ f , g ∈ D [ [ X ] ] , g ≠ 0 , and c ( f ) ⊆ c ( g ) ∗ } . Among other things, we show that Kr ( ( D , ∗ ) ) is a Bezout domain, D is completely integrally closed, the v -operation on D is a c-star operation, and Kr ( ( D , v ) ) is a completely integrally closed Bezout domain. We also show that if V is a rank-one valuation domain, then the v -operation on V is a c-star operation, Kr ( ( V , v ) ) is a rank-one valuation domain, and Kr ( ( V , v ) ) is a DVR if and only if V is a DVR. Using this result, we show that if D is a generalized Krull domain, then Kr ( ( D , v ) ) is a one-dimensional generalized Krull domain.

Factorization theory in commutative monoids

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

Class group of the ring of invariants of an exponential map on an affine normal domain

Let k be a field and let B be an affine normal domain over k. Let $$\phi $$ be a non-trivial exponential map on B and let $$A = B^{\phi }$$ be the ring of $$\phi $$ -invariants. Since A is factorially closed in B, $$A = K \cap B$$ where K denotes the field of fractions of A. Hence A is a Krull domain. We investigate here a relation between the class group $$\mathrm{Cl}(A)$$ of A and the class group $$\mathrm{Cl}(B)$$ of B. In this direction, we give a sufficient condition for an injective group homomorphism from $$\mathrm{Cl}(A)$$ to $$\mathrm{Cl}(B)$$ . We also give an example to show that $$\mathrm{Cl}(A)$$ may not be realized as a subgroup of $$\mathrm{Cl}(B)$$ .

Almost Krull domains and their rings of integer–valued polynomials

We investigate some ring theoretic properties of almost Krull domains. By using the language of star operations, we shed new light on the work of Pirtle on almost Krull domains. That allows us to give a new characterization of these domains. On the other hand, we show that Int(D) issued from an almost Krull domain is D–locally free and we give a complete characterization of when Int(D) is an almost Krull domain.

Factoriality and class groups of cluster algebras

Locally acyclic cluster algebras are Krull domains. Hence their factorization theory is determined by their (divisor) class group and the set of classes containing height-1 prime ideals. Motivated by this, we investigate class groups of cluster algebras. We show that any cluster algebra that is a Krull domain has a finitely generated free abelian class group, and that every class contains infinitely many height-1 prime ideals. For a cluster algebra associated to an acyclic seed, we give an explicit description of the class group in terms of the initial exchange matrix. As a corollary, we reprove and extend a classification of factoriality for cluster algebras of Dynkin type. In the acyclic case, we prove the sufficiency of necessary conditions for factoriality given by Geiss–Leclerc–Schröer.

Structural properties of subadditive families with applications to factorization theory

Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets $\mathscr U_k$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large $k$, namely, $H$ satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem. More precisely, we show that, under mild assumptions on $H$, not only does the Structure Theorem for Unions hold, but there also exists $\mu\in{\bf N}^+$ such that, for every $M\in{\bf N}$, the sequences $$ \bigl((\mathscr U_k-\inf\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} \quad\text{and}\quad \bigl((\sup\mathscr U_k-\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} $$ are $\mu$-periodic from some point on. The result applies, e.g., to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids. Large parts of the proofs are worked out in a "purely additive model", by inquiring into the properties of what we call a subadditive family, i.e., a collection $\mathscr L$ of subsets of $\bf N$ such that, for all $L_1,L_2\in\mathscr L$, there is $L\in\mathscr L$ with $L_1+L_2\subseteq L$.

Perinormality in pullbacks

In previous work, we introduced the notion of perinormality and showed how it lies in the greater context of commutative algebra, fitting as it does between the class of Krull domains and the class of seminormal ($R_1$) domains. Here we develop the concept further, using several pullback (i.e., gluing) constructions that yield perinormal domains. In doing so, we introduce the concepts of relative perinormality and fragility for ring extensions, which we believe to be of independent interest.

On elasticities of locally finitely generated monoids

Let H be a commutative and cancellative monoid. The elasticity ρ(a) of a non-unit a∈H is the supremum of m/n over all m,n for which there are factorizations of the form a=u1⋅…⋅um=v1⋅…⋅vn, where all ui and vj are irreducibles. The elasticity ρ(H) of H is the supremum over all ρ(a). We establish a characterization, valid for finitely generated monoids, when every rational number q with 1<q<ρ(H) can be realized as the elasticity of some element a∈H. Furthermore, we derive results of a similar flavor for locally finitely generated monoids (they include all Krull domains and orders in Dedekind domains satisfying certain algebraic finiteness conditions) and for weakly Krull domains.

On the structure of ⋆-power conductor domains

Let D be an integral domain and a star operation defined on D. We say that D is a -power conductor domain (-PCD) if for each pair and for each positive integer n we have We study -PCDs and characterize them as root closed domains satisfying for all nonzero a, b and all natural numbers . From this it follows easily that Prüfer domains are d-PCDs (where d denotes the trivial star operation), and v-domains (e.g. Krull domains) are v-PCDs. We also consider when a -PCD is completely integrally closed, and this leads to new characterizations of Krull domains. In particular, we show that a Noetherian domain is a Krull domain if and only if it is a w-PCD.

A homological characterization of Krull domains II

In this article, we introduce a new type of projective modules, called the weak w-projective module. By using this type of modules, we give a homological characterization of Krull domains. More precisely, it is shown that an integral domain is a Krull domain if and only if every submodule of a projective module is weak w-projective.

Sets of arithmetical invariants in transfer Krull monoids

Transfer Krull monoids are a recently introduced class of monoids and include the multiplicative monoids of all commutative Krull domains as well as of wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between 1 and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.

Long Sets of Lengths With Maximal Elasticity

AbstractWe introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid H, let Δρ(H) be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity ρ(H). We study Δρ(H) for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

Nearly Krull domains and nearly Prufer $v$-multiplication domains

In this paper, we introduce some new concepts of almost factoriality of integral domains. More precisely, we investigate nearly Krull domains, nearly Prufer $v$-multiplication domains and some related integral domains.

Demazure Construction for ℤn-Graded Krull Domains

For a Mori dream space X, the Cox ring Cox(X) is a Noetherian $\mathbb {Z}^{n}$ -graded normal domain for some n > 0. Let C(Cox(X)) be the cone (in $\mathbb {R}^{n}$ ) which is spanned by the vectors $\boldsymbol {a} \in \mathbb {Z}^{n}$ such that Cox(X)a≠ 0. Then, C(Cox(X)) is decomposed into a union of chambers. Berchtold and Hausen (Michigan Math. J., 54(3) 483–515: 2006) proved the existence of such decompositions for affine integral domains over an algebraically closed field. We shall give an elementary algebraic proof to this result in the case where the homogeneous component of degree 0 is a field. Using such decompositions, we develop the Demazure construction for $\mathbb {Z}^{n}$ -graded Krull domains. That is, under an assumption, we show that a $\mathbb {Z}^{n}$ -graded Krull domain is isomorphic to the multi-section ring R(X;D1,…, Dn) for certain normal projective variety X and $\mathbb {Q}$ -divisors D1, …, Dn on X.

Krull dimension of power series rings over non-SFT domains

A ring D is called an SFT ring if for each ideal I of D, there exist a finitely generated ideal J of D with J⊆I and a positive integer k such that ak∈J for all a∈I. For a cardinal number α and a ring D, we say that dim⁡D≥α if D has a chain of prime ideals with length ≥α. Arnold showed that if D is a non-SFT ring, then dim⁡D〚X〛≥ℵ0. Let C be the class of non-SFT domains. The class C includes the class of finite-dimensional nondiscrete valuation domains, the class of non-Noetherian almost Dedekind domains, the class of completely integrally closed domains that are not Krull domains, the class of integral domains with non-Noetherian prime spectrum, and the class of integral domains with a nonzero proper idempotent ideal. The ring of algebraic integers, the ring of integer-valued polynomials on Z, and the ring of entire functions are also members of the class C. In this paper we prove that dim⁡D〚X〛≥2ℵ1 for every D∈C and that under the continuum hypothesis 2ℵ1 is the greatest lower bound of dim⁡D〚X〛 for D∈C. On the other hand, there exists a (finite-dimensional) SFT domain D such that dim⁡D〚X〛≥2ℵ1.

A HOMOLOGICAL CHARACTERIZATION OF KRULL DOMAINS

In this article, we introduce a new type of projective modules, called the weak w-projective module. By using this type of modules, we give a homological characterization of Krull domains. More pre...