Related papers: Essential domains and two conjectures in dimension…

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…

Let $D$ be an integral domain with quotient field $K,$ $E$ a subset of $K$ and $X$ an indeterminate over $K$. The set $\mathrm{Int}(E,D):=\{f\in K[X];\; f(E)\subseteq D\}$, of integer-valued polynomials on $E$ over $D$, is known to be an…

Let $D$ be an integral domain with quotient field $K$ and $E$ a subset of $K$. The \textit{ring of integer-valued rational functions on} $E$ is defined as $$\mathrm{int}_R(E,D):=\lbrace \varphi \in K(X);\; \varphi(E)\subseteq D\rbrace.$$…

It is well-known that for any commutative unitary ring $\mathbf{R}$, the Serre conjecture ring $\mathbf{R}\langle X \rangle$, i.e., the localization of the univariate polynomial ring $\mathbf{R}[X]$ at monic polynomials, is a B\'ezout…

In this paper we develop the theory of essential dimension of group schemes over an integral base. Shortly we concentrate over a local base. As a consequence of our theory we give a result of invariance of the essential dimension over a…

We study K_2 of one-dimensional local domains over a field of characteristic 0, introduce a conjecture, and show that this conjecture implies Geller's conjecture. We also show that Berger's conjecture implies Geller's conjecture, and hence…

We introduce the notion of Krull super-dimension of a super-commutative super-ring. This notion is used to describe regular super-rings and calculate Krull super-dimensions of completions of super-rings. Moreover, we use this notion to…

We investigate the differential Krull dimension of differential polynomials over a differential ring. We prove a differential analogue of Jaffard's Special Chain Theorem and show that differential polynomial extensions of certain classes of…

This is an expository paper in which it is proved that, for every infinite field ${\mathbf{F}}$, the polynomial ring ${\mathbf{F}}[t_1,\ldots, t_n]$ has Krull dimension $n$. The proof uses only "high school algebra" and the rudiments of…

Let $A$ be a commutative arithmetical ring. The ring $A$ has Krull dimension if and only if every factor ring of $A$ is finite-dimensional and does not have idempotent proper essential ideals. The study is supported by Russian Science…

We give a proof of the Greene-Krantz conjecture on convex domains in $\CC^2$. Curiously, the proof technique depends on subelliptic estimates for the $\bar{\partial}$ problem.

Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over $D$. We introduce and…

We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espa\~nol and the authors. We show that the notion of Krull…

We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…

We look for generalizations of the Brown-Douglas-Fillmore essential codimension result, leading to interesting local uniqueness theorems in $KK$-theory. We also study the structure of Paschke dual algebras.

By analysing the structure of the associated graded ring with respect to certain filtrations, we deduce a number of good properties of iterated local skew power series rings over appropriate base rings. In particular, we calculate the Krull…

This paper deals with Bouvier's conjecture which sustains that finite-dimensional non-Noetherian Krull domains need not be Jaffard

This is an extended introduction to discrete valuation rings and Dedekind domains. Some natural generalizations of Dedekind domains are also (briefly) discussed including "almost Dedekind domains", Pr\"ufer domains, Krull domains, and…

Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of…