Related papers: On some local rings

We study properties of the Golomb topology on polynomial rings over fields, in particular trying to determine conditions under which two such spaces are not homeomorphic. We show that if $K$ is an algebraic extension of a finite field and…

We study the interplay of the Golomb topology and the algebraic structure in polynomial rings $K[X]$ over a field $K$. In particular, we focus on infinite fields $K$ of positive characteristic such that the set of irreducible polynomials of…

We consider differential rings of the form (K[x; y];D), where K is an algebraically closed field of characteristic zero and D : K[x; y] \to K[x; y] is a K-derivation. We study the Automorphism Group of such a ring and give criteria for…

Let $\mathcal{O}_2$ and $\mathcal{O}'_2$ be two distinct finite local rings of length two with residue field of characteristic $p$. Let $\mathbb{G}(\mathcal{O}_2)$ and $\mathbb{G}(\mathcal{O}'_2)$, be the group of points of any reductive…

A criterion is given which assures that two p-divisible groups X and Y over an algebraically closed field of characteristic p are isomorphic when their p-kernels X[p] and Y[p] are isomorphic.

In this article it is determined which integral reflection representations of the symmetric groups and the primitive complex reflection groups of degree $2$ have rings of invariants which are isomorphic to polynomial rings.

We associate to every quandle $X$ and an associative ring with unity $\mathbf{k}$, a nonassociative ring $\mathbf{k}[X]$ following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner…

Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if…

In this paper we prove that if $R$ is a commutative, reduced, local ring, then $R$ is Hopfian if and only if the ring $R[x]$ is Hopfian. This answers a question of Varadarajan, in the case when $R$ is a reduced local ring. We provide…

Let $K$ be an algebrically closed field and let $n\geq 1$. If $P\in K[X]=K[X_1,\ldots,X_n]$, $P\neq 0$, we denote by $I(P)$ the support of $P$, which is the finite subset of $\mathbb N^n$ such that $P=\sum_{i\in I(P)}a_iX^i$ with $a_i\in…

Let $K$ be the field of Laurent series with complex coefficients, let $\mathcal{R}$ be the inverse limit of the standard-graded polynomial rings $K[x_1, \ldots, x_n]$, and let $\mathcal{R}^{\flat}$ be the subring of $\mathcal{R}$ consisting…

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 $\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…

In this paper, we provide a simple proof for the fact that two simplicial complexes are isomorphic if and only if their associated Stanley-Reisner rings, or their associated facet rings are isomorphic as $K$-algebras. As a consequence, we…

We answer a question by Shestakov on the Jacobson radical in differential polynomial rings. We show that if R is a locally nilpotent ring with a derivation D then R[X;D] need not be Jacobson radical. We also show that J(R[X;D])\cap R is a…

Let $K$ be a local field with residue characteristic $p$ and let $L/K$ be a totally ramified extension of degree $p^k$. In this paper we show that if $L/K$ has only two distinct indices of inseparability then there exists a uniformizer…

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial…

We will prove that if $G$ and $H$ are modules over a principal ideal domain $R$ such that the endomorphism rings $\mathrm{End}_R(R\oplus G)$ and $\mathrm{End}_R(R\oplus H)$ are isomorphic then $G\cong H$. Conversely, if $R$ is a Dedekind…

We study iterated differential polynomial rings over a locally nilpotent ring and show that a large class of such rings are Behrens radical. This extends results of Chebotar and Chen et al.

Let $K$ be a number field with ring of integers $\mathbb{Z}_K$. We prove two asymptotic formulas connected with the distribution of irreducible elements in $\mathbb{Z}_K$. First, we estimate the maximum number of nonassociated irreducibles…