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In this article, a new semistar operation, called the q-operation, on a commutative ring R is introduced in terms of the ring Q0(R) of finite fractions. It is defined as the map q:Fq(R)→Fq(R) by A↦...

Given an ideal a in A [ x 1 , … , x n ] where A is a Noetherian integral domain, we propose an approach to compute the Krull dimension of A [ x 1 , … , x n ] / a , when the residue class ring is a free A-module. When A is a field, the Krull dimension of A [ x 1 , … , x n ] / a has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. For a Noetherian integral domain A we introduce the notion of combinatorial dimension of A [ x 1 , … , x n ] / a and give a Grobner basis method to compute it for residue class rings that have a free A-module representation w.r.t. a lexicographic ordering. For such A-algebras, we derive a relation between Krull dimension and combinatorial dimension of A [ x 1 , … , x n ] / a . An immediate application of this relation is that it gives a uniform method, the first of its kind, to compute the dimension of A [ x 1 , … , x n ] / a without having to consider individual properties of the ideal. For A-algebras that have a free A-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Grobner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A-algebras with a free A-module representation w.r.t. a degree compatible ordering as well.

We propose a new version of Noether normalization theorem over Bezout domains of Krull dimension one (the ring Z of integers as main example). We also show that one can avoid branching when computing dynamical Grobner bases over a Bezout domain of Krull dimension one.

We give an exposition of a recent result of M. Kapovich on the cardinal Krull dimension of the ring of holomorphic functions on a connected C -manifold. By reducing to the one-dimensional case we give a stronger lower bound for Stein manifolds.

We prove that for any valuation ring R of Krull dimension ≤1 or any commutative local ring R with Krull dimension 0 and n≥3, the special linear group SLn(R[x]) coincides with the group of elementary matrices En(R[x]), and for an arbitrary arithmetical ring R of Krull dimension ≤1 and n≥3, the group SLn(R[x])=SLn(R)⋅En(R[x]). These give partial solutions to two conjectures of Yengui.

In this paper, we concern the Principal Ideal Theorem (PIT) for w-Noetherian rings. Let R be a w-Noetherian ring and a be a nonzero nonunit element of R. If p is a prime ideal of R minimal over (a), then ht p ≦ 1.

It is well known that the equivalence of injectivity and divisibility characterizes the Dedekind domains. In this note, we shed more light on this problem and show that a similar property characterizes Krull domains and π-domains.

In this article, we give a complete characterization of when the (composite) semigroup ring is a generalized Krull domain.

Noncommutative Krull domains that are determined by submonoids of torsion-free nilpotent groups are investigated. A complete description is given in case the group $G$ is nilpotent of class two and its abelianisation is torsion-free and satisfies the ascending chain condition on cyclic subgroups. The result corrects and extends an earlier result by the authors to the case that $G$ is not necessarily finitely generated and yields a class of non-Noetherian algebras that have a nice arithmetical structure.

Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property () if for each nonzero element , each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property () if and only if R has no R-sequence of length 2; the "if" assertion fails in general for non-domain rings R. Each treed domain has property (), but the converse is false.

Let $P$ be a finitely generated ideal of a commutative ring $R$. Krull's Principal Ideal Theorem states that if $R$ is Noetherian and $P$ is minimal over a principal ideal of $R$, then $P$ has height at most one. Straightforward examples show that this assertion fails if $R$ is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem., 15 pages, to appear in Math. Proc. Cambridge Philos. Soc

This corrigendum is written to correct some parts of the paper "On density theorems for rings of Krull type with zero divisors". The proofs of Proposition 2.4 and Proposition 4.3 are incorrect and the current note makes the appropriate corrections.

We determine the Krull dimension of the module category of finite dimensional tame generalized multicoil algebras over an algebraically closed field, which are domestic.

We prove that the Krull dimension of the ring of holomorphic functions of a connected complex manifold is at least continuum if it is positive., Comment: 6 pages. An error pointed out by Pete Clark is corrected. The stronger statement about the Krull dimension at least continuum is proven

Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, sets of lengths of transfer Krull monoids and of weakly Krull monoids satisfy the Structure Theorem for Sets of Lengths. In spite of this common feature we demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid.

We define and study two generalizations of the Krull dimension for rings, which can assume cardinal number values of arbitrary size. The first, which we call the "cardinal Krull dimension," is the supremum of the cardinalities of chains of prime ideals in the ring. The second, which we call the "strong cardinal Krull dimension," is a slight strengthening of the first. Our main objective is to address the following question: for which cardinal pairs (K,L) does there exist a ring of cardinality K and (strong) cardinal Krull dimension L? Relying on results from the literature, we answer this question completely in the case where K>L or K=L. We also give several constructions, utilizing valuation rings, polynomial rings, and Leavitt path algebras, of rings having cardinality K and (strong) cardinal Krull dimension L>K. The exact values of K and L that occur in this situation depend on set-theoretic assumptions., 22 pages

Using the concept of prime submodule defined by Raggi et al. in [16], for M ∈ R-Mod we define the concept of classical Krull dimension relative to a hereditary torsion theory τ ∈M-tors. We prove that if M is progenerator in σ[M], τ ∈M-tors such that M has τ-Krull dimension then cl.K τdim (M) ≤ k τ(M). Also we show that if M is noetherian, τ-fully bounded, progenerator of σ[M], and M ∈ 𝔽τ, then cl·K τdim (M) = k τ(M).

In the early 1990's, Lionel Schwartz gave a lovely characterization of the Krull filtration of U, the category of unstable modules over the mod p Steenrod algebra. Soon after, this filtration was used by the author as an organizational tool in posing and studying some topological nonrealization conjectures. In recent years the Krull filtration of U has been similarly used by Castellana, Crespo, and Scherer in their study of H--spaces with finiteness conditions, and Gaudens and Schwartz have given a proof of some of my conjectures. In light of these topological applications, it seems timely to better expose the algebraic properties of the Krull filtration., 21 pages

In this paper, we introduce the notion of upper (down) set in BL-algebras and extended upper sets of BL-algebras. Also, the notions of Krull dimension and regular sequence in BL-algebras are introduced and studied in detail.

We introduce a naive notion of a system of parameters for a homologically finite complex over a commutative noetherian local ring, and compare it to the system of parameters defined by Christensen. We show that these notions differ in general, but that they agree when the complex in question is a DG R-algebra. In this case we also show that the Krull dimension defined in terms of the lengths of such systems of parameters agrees with Krull dimensions defined in terms of certain chains of prime ideals., 8 pages, v.2 has minor editorial changes. to appear in Arch. Math. (Basel)

We give a point-free and constructively meaningful characterization of the notion of codimension for ideals of a commutative ring, which with classical logic and the axiom of choice is equivalent to the customary one. This characterization is based on notions from formal topology, and upon the elementary characterization of Krull dimension provided earlier by Coquand, Lombardi, and Roy. Among other things, we use this notion to prove Krullʼs principal ideal theorem in the context of constructive algebra.

It is known that in a right Noetherian ring with finite Krull dimension, the idealiser of a finite intersection of maximal right ideals is right Noetherian with finite Krull dimension. This result is generalised to a finite intersection of right ideals for each of which, essentially, the corresponding factor module is critical and has an endomorphism ring given by a right Ore domain. It is then shown that in the ring of differential operators on a regular affine domain, this result applies to the idealiser of a right ideal generated by the prime ideal of a smooth sub-variety of arbitrary co-dimension, and to the idealiser of a finite intersection of such right ideals, provided the corresponding primes are pairwise co-maximal and of equal height. An application is given to prove that the ring of differential operators on a quotient variety formed by glueing together the images of a smooth subvariety under a finite group action is right Noetherian.

for example, because of their connections with number theory and arithmetic geometry; see the paper by Coates, Schneider and Sujatha [4] for more details. When G is a uniform pro-p group, ΛG is a concrete example of a complete local Noetherian ring (noncommutative, in general) with good homological properties: it is known that ΛG has finite global dimension and is an Auslander regular ring. Thus, ΛG falls into the class of rings studied by Brown, Hajarnavis and MacEacharn in [1]. There they consider various properties of Noetherian rings R of finite global dimension, including the Krull(–Gabriel–Rentschler) dimension K(R)—a module-theoretic dimension which measures how far R is from being Artinian. They also posed the following question

For any prime p, we show that the mod p Steenrod algebra (a local ring with nil maximal ideal) has infinite little Krull dimension. This contrasts sharply with the case of a commutative (or noetherian) local ring with nil maximal ideal which must have little Krull dimension equal to 0. Also, we show that the Steenrod algebra has no Krull dimension, classical Krull dimension, or Gabriel dimension.

Let R be a graded Noetherian domain and A a graded Krull overring of R. We show that if h-dimR 2, then A is a graded Noetherian domain with h-dimA 2. This is a generalization of the well-known theorem that a Krull overring of a Noetherian domain with dimension 2 is also a Noetherian domain with dimension 2.

In this article we shall prove the following result, which is a generalization of the Lichtenbaum–Hartshorne Vanishing Theorem (See [3, Theorem 8.2.1]). Let (R, 𝔪) be a catenary Noetherian local ring, I an ideal of R and M be a nonzero finitely generated R-module of dimension n. Then the following conditions are equivalent: (i) and (ii) there exists a prime ideal 𝔭 in AsshR(M) such that Rad(𝔭 +I) = 𝔪.

It is proven that if R is a commutative Bezout ring of Krull dimension 1, with stable range 2, then R is an elementary divisor ring.

Let R be a commutative Noetherian ring and σ an automorphism of R. Let K be a commutative Noetherian ring, which is also an algebra over \({\mathbb{Q}}\) , and δ a derivation of K (\({\mathbb{Q}}\) is the field of rational numbers). Consider the rings R[x; σ], R[x, x −1; σ] and K[x; δ]. We show that if I is a two-sided ideal in any one of these rings, then the left and right Krull dimensions of I coincide.

We classify tilting classes over regular rings R of Krull dimension two. They are parametrized by the set of all pairs ( X , Y ) such that Ass R R ⊆ X ⊆ Spec ( R ) , Y consists of maximal ideals of height 2, and Y contains all the maximal ideals of height 2 that contain some element of X ∖ Ass R R . For R local, we also classify the corresponding infinitely generated tilting modules.

The algebra of basic covers of a graph G, denoted by \A(G), was introduced by Juergen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite then \A(G) is a homogeneous algebra with straightening laws and thus is Koszul. Furthermore, we compute the Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs., Comment: 23 pages

The concept of ideals play an important role in Hilbert Algebras in BCK-algebras, so here we will give some relationships between ideal, fuzzy ideal, commutative fuzzy ideal and weak ideal.

In this note we show that the quotient field of a domain which is Henselian with respect to a nontrivial ideal is a large field, and give some applications of this fact, using a specialization theorem for ramified covers of the line over (generalized) Krull fields.

Let M be a Krull monoid. Then every element of M may be written as a finite product of irreducible elements. If for every a ∈ M, each two factorizations of a have the same number of irreducible elements, then M is called half-factorial. Using a property of element exponentiation, we provide a new characterization of half-factoriality, valid for all Krull monoids whose class group has torsion-free rank at most one.

Let R be an integral domain. We say that R is a star-domain if R has at least a height one prime ideal and if for each height one prime ideal P of R, R satisfies the acc on P-principal ideals (i.e., ideals of the form aP, a ∈ R). We prove that if R is an APVD with nonzero finite Krull dimension, then the power series ring R[[X]] has finite Krull dimension if and only if R is a residually star-domain (i.e., for each nonmaximal prime ideal P of R, R/P is a star-domain) if and only if R[[X]] is catenarian.

In this survey paper we present some results relating the Goldie dimension, dual Krull dimension and subdirect irreducibility in modules, torsion theories, Grothendieck categories and lattices. Our interest in studying this topic is rooted in a nice module theoretical result of Carl Faith [Commun. Algebra27 (1999), 1807–1810], characterizing Noetherian modules M by means of the finiteness of the Goldie dimension of all its quotient modules and the ACC on its subdirectly irreducible submodules. Thus, we extend his result in a dual Krull dimension setting and consider its dualization, not only in modules, but also in upper continuous modular lattices, with applications to torsion theories and Grothendieck categories.

Extensions of the notion of a class group and a block monoid associated to a Krull monoid with torsion class group are introduced and investigated. Instead of assigning to a Krull monoid only one abelian group (the class group) and one monoid of zero-sum sequences (the block monoid), we assign to it a recursively defined family of abelian groups, the first being the class group, and do alike for the block monoid. These investigations are motivated by the aim of gaining a more detailed understanding of the arithmetic of Krull monoids, including Dedekind and Krull domains, both from a technical and conceptual point of view. To illustrate our method, some first arithmetical applications are presented.

We resolve an open problem in commutative algebra and Field Arithmetic, posed by Jarden -- Let R be a generalized Krull domain. Is the ring R[[X]] of formal power series over R a generalized Krull domain? We show that the answer is negative., 7 pages

In this paper we consider a partial action α of a polycyclic by finite group G on a ring R. We prove that if R is right noetherian, then the partial skew group ring R ⋆ αG is also right noetherian. Extending the methods of Passman in Passman (Trans Am Math Soc 301:737–759, 1987), we obtain a description of the prime spectrum of R ⋆ αG. The results obtained are applied to get bounds for the Krull dimension and the classical Krull dimension of R ⋆ αG.

For a commutative ring R with identity, dim R shall stand for the Krull dimension of R . It is known that dim R [ x ] ≤ 2 dim R + 1 . We show that this does not hold for the power series extensions. Using mixed extensions, we construct an example of a finite-dimensional integral domain R such that 2 dim R + 1 dim R 〚 x 〛 ∞ . Let D be a finite-dimensional SFT Prufer domain and D [ x 1 〛 ⋯ [ x n 〛 be a mixed extension. According to Arnold, dim D [ x 1 , … , x n ] = dim D + n and dim D 〚 x 1 , … , x n 〛 = n dim D + 1 . We generalize Arnold’s result by showing that dim D [ x 1 〛 ⋯ [ x n 〛 = n dim D + 1 provided that there is at least one power series extension. In particular, if R = D [ x 1 , ⋯ , x n − 1 ] and dim D = d > 2 ( n − 1 ) / ( n − 2 ) , then dim R 〚 x 〛 = d n + 1 > 2 dim R + 1 . This is an answer to the question of Coykendall and Gilmer.

We give a constructive proof of the fact that finitely generated projective modules over a polynomial ring with coefficients in a Prufer domain R with Krull dimension ≤ 1 are extended from R. In particular, we obtain constructively that finitely generated projective R[X1, …, Xn ]-modules, where R is a Bezout domain with Krull dimension ≤ 1, are free. Our proof is essentially based on a dynamical method for decreasing the Krull dimension and a constructive rereading of the original proof given by Maroscia and Brewer & Costa. Moreover, we obtain a simple constructive proof of a result due to Lequain and Simis stating that finitely generated modules over R[X1, …, Xn ], n ≥ 2, are extended from R if and only if this holds for n = 1, where R is an arithmetical ring with finite Krull dimension (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study skew inverse power series extensions R[[y− 1; τ, δ]], where R is a noetherian ring equipped with an automorphism τ and a τ-derivation δ. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete, local, noetherian, Auslander regular domains whose right Krull dimension, global dimension, and classical Krull dimension are all equal to 2n.

Noetherian rings with Krull dimension one are shown to have closed left ideals in the J-adic topology. The radical of these rings also satisfies the AR property.

We constructively prove that for any ring R with Krull dimension ⩽d, the ring R〈X〉 locally behaves like the ring R(X) or a localization of a polynomial ring of type (S−1R)[X] with S a multiplicative subset of R such that the Krull dimension of S−1R is ⩽d−1. As an application, we give a simple and constructive proof of the Lequain–Simis Induction Theorem which is an important variation of the Quillen Induction Theorem.

We prove constructively that for any ring R of Krull dimension ⩽1 and n ⩾ 3 , the group E n ( R [ X ] ) acts transitively on Um n ( R [ X ] ) . In particular, we obtain that for any ring R with Krull dimension ⩽1, all finitely generated stably free modules over R [ X ] are free. This settles the long-standing Hermite ring conjecture for rings of Krull dimension ⩽1.

Let D be a Noetherian domain of Krull dimension 2, and let H⊆R be integrally closed overrings of D. We examine when H can be represented in the form H=(⋂V∈ΣV)∩R, with Σ a Noetherian subspace of the Zariski–Riemann space of the quotient field of D. We characterize also the special case in which Σ can be chosen to be a finite character collection of valuation overrings of D.

Let D be a two-dimensional Noetherian domain, let R be an overring of D, and let Σ and Γ be collections of valuation overrings of D. We consider circumstances under which ( ⋂ V ∈ Σ V ) ∩ R = ( ⋂ W ∈ Γ W ) ∩ R implies that Σ = Γ . We show that if R is integrally closed, these representations are “strongly” irredundant, and every member of Σ ∪ Γ has Krull dimension 2, then Σ = Γ . If in addition Σ and Γ are Noetherian subspaces of the Zariski–Riemann space of the quotient field of D (e.g. if Σ and Γ have finite character), then the restriction that the members of Σ ∪ Γ have Krull dimension 2 can be omitted. An example shows that these results do not extend to overrings of three-dimensional Noetherian domains.

Let R be a commutative Noetherian ring of Krull dimension d , and let a be an ideal of R . In this paper, we will study the strong cotorsioness and the Gorenstein injectivity of the section functor Γ a ( − ) in local cohomology. As applications, we will find new characterizations for Gorenstein and regular local rings. We also study the effect of the section functors Γ a ( − ) and the functors H a d ( − ) on the Auslander and Bass classes.