On a radical extension of the field of rational functions in several variables

The ideals of an ideal extension

Given a unital associative ring S and a subring R, we say that S is an ideal (or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum is direct. In this note we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These …

Classification of bilinear maps with radical of codimension 2

Let ${\mathbb V}$ be an $n$-dimensional linear space over an algebraically closed base field. We provide a classification, up to equivalence, of all of the bilinear maps $f:{\mathbb V} \times {\mathbb V} \to {\mathbb V}$ such that $dim(rad(f)) =n-2$. This is equivalent to give a complete classification (up to isomorphism) of all $n$-dimensional algebras with annihilator of dimension $n-2$ or, in other words, a classification of the annihilator extensions of all $2$-dimensional algebras.

Ore扩张的Jacobson根

Jacobson radicals of Ore extensions

本文研究Ore扩张$R[x;\sigma,D]$的Jacobson根，给出特征零下$q$-斜导子情形的结构公式，并证明在PI环或正特征局部幂零导子等条件下其与系数环的交为诣零理想。

单侧理想的整依赖，第一部分：基础与经典应用

Integral Dependence over One-Sided Ideals, Part I: Foundations and Classical Applications

本文引入单侧理想的整性与完全整性概念及相应根构造，建立一般研究框架，并将Jacobson根相关经典结果推广至单侧情形，证明左Artin环与代数代数中左理想的Jacobson根整性及分次性，推广了Amitsur定理。

有理向量场的李维利可积性：代数扩张的情况

Liouvillian integrability of rational vector fields: The case of algebraic extensions

本文研究了有理向量场在代数扩张下的李维利可积性，证明了存在例外向量场，并在三维空间中给出了显式构造。

On rational extension of Heisenberg algebra

Construction of rational extension for Heisenberg algebra with one pair of generators is discussed.

Rational Extensions of C(X) via Hausdorff Continuous Functions

The ring operations and the metric on $C(X)$ are extended to the set $\mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $\mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions.

B*-algebras over ultranormed fields

This article is devoted to the investigation of $B^*$-algebras, dual and annihilator ultranormed algebras. Their structure is studied in the paper. Extensions of algebras and fields are considered and using them core radicals and radicals are investigated. Moreover, for this purpose also $*$-algebras and finely regular algebras are studied. Relations with operator theory and realizations of these algebras by operator algebras are outlined.

Galois Groups in Rational Conformal Field Theory

It was established before that fusion rings in a rational conformal field theory (RCFT) can be described as rings of polynomials, with integer coefficients, modulo some relations. We use the Galois group of these relations to obtain a local set of equation for the points of the fusion variety. These equations are sufficient to classify all the RCFT, Galois group by Galois group. It is shown that the Galois group is equivalent to the pseudo RCFT group. We prove that the Galois groups encountered in RCFT are all abelian, implying solvability by radicals of the modular matrix.

On Extensions of Rational Modules

We investigate when the categories of all rational $A$-modules and of finite dimensional rational modules are closed under extensions inside the category of $C^*$-modules, where $C^*$ is the cofinite topological completion of $A$. We give a complete characterization of these two properties, in terms of a topological and a homological condition. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes …

Extensions of almost faithful prime ideals in virtually nilpotent mod-$p$ Iwasawa algebras

Let $G$ be a nilpotent-by-finite compact $p$-adic analytic group for some $p>2$, and $H = \mathbf{FN}_p(G)$ its finite-by-(nilpotent $p$-valuable) radical. Fix a finite field $k$ of characteristic $p$, and write $kG$ for the completed group ring of $G$ over $k$. We show that almost faithful $G$-stable prime ideals $P$ of $kH$ extend to prime ideals $PkG$ of $kG$.