Decomposition of the Witt-Burnside ring and Burnside ring of an abelian profinite group

Let G, H be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers. The aim of this paper is to establish an isomorphism of functors , where denotes the q-deformed Witt-Burnside ring functor of G introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. To do this, we first establish an isomorphism of functors , where denotes the q-deformed Burnside ring functor of G which was also introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. As an application, we derive a pseudo-multiplicative property of the q-Möbius function associated to the lattice of open subgroups of the direct sum of G and H.     

Overrings of two-dimensional Noetherian domains representable by Noetherian spaces of valuation rings

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.     

The Green rings of pointed tensor categories of finite type

In this paper, we compute the Clebsch–Gordan formulae and the Green rings of connected pointed tensor categories of finite type.     

Intermediary rings in normal pairs

We establish in this paper a result that gives the number of intermediary rings between R and S where (R,S) is a normal pair of rings. This result answers in particular a question which was left open in [A. Jaballah, Finiteness of the set of intermediary rings in a normal pair, Saintama Math. J. 17 (1999) 59-61]. Further applications are also given.     

Almost slender rings and compact rings

Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring $k^{\mathbb{N}}/k^{(\mathbb{N})}$ over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras $R^{\mathbb{N}}\to R$ vanishing on $R^{(\mathbb{N})}$.     

On quasi divisor theories and systems of valuations

We study the relationship between divisor theories and systems of valuations, and characterize monoids with quasi divisor theories of finite character by systems of essential valuations. Throughout, we avoid ideal theory but use divisor theoretical methods.     

A Tate cohomology sequence for generalized Burnside rings

We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If D is a restriction functor for a finite group G, then the mark morphism φ:D₊→D⁺ is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G) after composing with a suitable isomorphism of D⁺. As a consequence, we obtain an exact sequence of Mackey functors

     

Universally catenarian and going-down pairs of rings

For a ring extension is called a universally catenarian pair if every domain , is universally catenarian. When R is a field it is shown that the only universally catenarian pairs are those satisfying . For several satisfactory results are given. The second purpose of this paper is to study going-down pairs (Definition 5.1). We characterize these pairs of rings and we establish a relationship between universally catenarian, going-down and residually algebraic pairs. Received: 1 July 1999; in final form: 5 June 2000 / Published online: 17 May 2001     

On the topology of valuation-theoretic representations of integrally closed domains

Let F be a field. For each nonempty subset X of the Zariski–Riemann space of valuation rings of F, let $A(X)={?}_{V\in X}V$ and $J(X)={?}_{V\in X}{\mathfrak{M}}_V$, where ${\mathfrak{M}}_V$ denotes the maximal ideal of V. We examine connections between topological features of X and the algebraic structure of the ring $A(X)$. We show that if $J(X)\ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of F, then there is a space Y of valuation rings of F that is perfect in the patch topology such that $A(X)=A(Y)$. If any countable subset of points is removed from Y, then the resulting set remains a representation of $A(X)$. Additionally, if F is a countable field, the set Y can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring $A(X)$ with specific focus on topological conditions that guarantee $A(X)$ is a Prüfer domain, a feature that is reflected in the Zariski–Riemann space when viewed as a locally ringed space. We also classify the rings $A(X)$ where X has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.     

On the class group and S-class group of formal power series rings

Let ${\text{Cl}}_t(A)$ denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is injective and if A is a regular UFD, then ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$; $[I]?[{(I.A?X?)}_t]$ is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an $s\in S$ and $a\in I$ such that $sI?aA?I$. The S-class group of A, S-${\text{Cl}}_t(A)$ is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-${\text{Cl}}_t(A)?S$-${\text{Cl}}_t(A?X?)$.     

An algorithm for computing the number of intermediary rings in normal pairs

The main purpose of this paper is to establish a result giving the number of intermediary rings between R and S when (R,S) is a normal pair of rings and to provide an algorithm to compute this number.     

Locally strong endomorphisms of paths

We determine the number of locally strong endomorphisms of directed and undirected paths—direction here is in the sense of a bipartite graph from one partition set to the other. This is done by the investigation of congruence classes, leading to the concept of a complete folding, which is used to characterize locally strong endomorphisms of paths. A congruence belongs to a locally strong endomorphism if and only if the number l of congruence classes divides the length of the original path and the points of the path are folded completely into the l classes, starting from 0 to l and then back to 0, then again back to l and so on. It turns out that for paths locally strong endomorphisms form a monoid if and only if the length of the path is prime or equal to 4 in the undirected case and in the directed case also if the length is 8. Finally some algebraic properties of these monoids are described.     

Groups with isomorphic Burnside rings

We prove that for some families of finite groups, the isomorphism class of the group is completely determined by its Burnside ring. Namely, we prove the following: if two finite simple groups have isomorphic Burnside rings, then the groups are isomorphic; if G is either Hamiltonian or abelian or a minimal simple group, and G is any finite group such that B(G) B(G), then G G.Received: 22 April 2004     

On the cohomology rings of small categories

Let C be a small category and R a commutative ring with identity. The cohomology ring of C with coefficients in R is defined as the cohomology ring of the topological realization of its nerve. First we give an example showing that this ring modulo nilpotents is not finitely generated in general, even when the category is finite EI. Then we study the relationship between the cohomology ring of a category and those of its subcategories and extensions. The main results generalize certain theorems in group cohomology theory.     

Almost maximally almost-periodic group topologies determined by T-sequences

A sequence {an} in a group G is a T-sequence if there is a Hausdorff group topology τ on G such that . In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a T-sequence, and investigate special sequences in the Prüfer groups Z(p^∞). We show that for p≠2, there is a Hausdorff group topology τ on Z(p^∞) that is determined by a T-sequence, which is close to being maximally almost-periodic—in other words, the von Neumann radical n(Z(p^∞),τ) is a non-trivial finite subgroup. In particular, n(n(Z(p^∞),τ))?n(Z(p^∞),τ). We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a T-sequence with non-trivial finite von Neumann radical.     

Note on non-D-rings

Abstract

Recall that an integral domain R is said to be a non-D-ring if there exists a non-constant polynomial f (X) in R[X] (called a uv-polynomial) such that f (a) is a unit of R for every a in R. In this note we generalize this notion to commutative rings (that are not necessarily integral domains) as follows: for a positive integer n, we say that R is an n-non-D-ring if there exists a polynomial f of degree n in R[X] such that f (a) is a unit of R for every a in R. We then investigate the properties of this notion in di?erent contexts of commutative rings.