Related papers: Polynomial Dedekind domains with finite residue fi…

We give a definition of a class of Dedekind domains which includes the rings of integers of global fields and give a proof that all rings in this class have finite ideal class group. We also prove that this class coincides with the class of…

Let $p\in\mathbb Z$ be a prime, $\overline{\mathbb Q_p}$ a fixed algebraic closure of the field of $p$-adic numbers and $\overline{\mathbb Z_p}$ the absolute integral closure of the ring of $p$-adic integers. Given a residually algebraic…

For a given family $(G_i)_{i \in \N}$ of finitely generated abelian groups, we construct a Dedekind domain $D$ having the following properties. \begin{enumerate} \item $\Pic(D) \cong \bigoplus_{i \in \N}G_i$. \item For each $i \in \N$,…

In this paper, as an extension of the integer case, we define polynomial functions over the residue class rings of Dedekind domains, and then we give canonical representations and counting formulas for such polynomial functions. In…

A classical result of Claborn states that every abelian group is the class group of a commutative Dedekind domain. Among noncommutative Dedekind prime rings, apart from PI rings, the simple Dedekind domains form a second important class. We…

For an integral domain $R$, the {\it ring of integer-valued polynomials} over $R$ consists of all polynomials $f(X) \in R[X]$ such that $f(R) \subseteq R$. An interesting case to study is when $R$ is a Dedekind domain, in particular when…

For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D'$ are Dedekind domains and there is an order isomorphism between $\mathcal{P}(D)$ and…

What are all rings $R$ for which $R^*$ (the group of invertible elements of $R$ under multiplication) is an elementary abelian $p$-group? We answer this question for finite-dimensional commutative $k$-algebras, finite commutative rings,…

We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field…

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain $(R,M)$ with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose…

Let R be a Dedekind domain, and let G be a simply connected Chevalley-Demazure group scheme of rank =>2. We prove that G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for any n=>1. This extends the corresponding results of A. Suslin and F.…

Analytic properties of function spaces over the real and the complex fields are different in some ways. This reflects in algebraic properties which are different at times and similar in some other respects. For instance, the ring of…

Let $F$ be a field, $p$ a prime number, $X$ an indeterminate over $F$, $D_n =F[X^{\frac{1}{p^n}}, X^{-\frac{1}{p^n}}]$ for each integer $n \geq 0$ and $D = \bigcup\limits_{n\in\mathbb{N}_0}D_n.$ Then $D$ is a one-dimensional B{\'e}zout…

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any…

Let F:K be a Galois extension of number fields and Q a prime ideal of O_F lying over the prime P of O_K. By analyzing the Q-adic closure of O_K in O_F we characterize those rings of integers O_K for which every residue class ring of…

Let $D$ be a domain with fraction field $K$, and let $M_n(D)$ be the ring of $n \times n$ matrices with entries in $D$. The ring of integer-valued polynomials on the matrix ring $M_n(D)$, denoted ${\rm Int}_K(M_n(D))$, consists of those…

Let $\S $ be an arbitrary subset of $R^n$ where $R$ is a domain with the field of fractions $\K$. Denote the ring of polynomials in $n$ variables over $\K$ by $\K[\x].$ The ring of integer-valued polynomials over $\S,$ denoted by…

This article investigates the properties of Dedekind superrings, invertible supermodules and projective supermodules within the $\mathbb{Z}_2$-graded framework. Rather than treating these entities as specialized instances of general…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

Let $D$ be an integrally closed domain with quotient field $K$ and $A$ a torsion-free $D$-algebra that is finitely generated as a $D$-module and such that $A\cap K=D$. We give a complete classification of those $D$ and $A$ for which the…