Related papers: Rings with Polynomial Identity and Centrally Essen…

We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover,…

The associative ring $R(P(t))=\mathbb C[t^{\pm1},u \,|\, u^2=P(t)]$, where $P(t)=\sum_{i=0}^na_it^i=\prod_{k=1}^n(t-\alpha_i)$ with $\alpha_i\in\mathbb C$ pairwise distinct, is the coordinate ring of a hyperelliptic curve. The Lie algebra…

After the language of module and theirs morphisms, this short course presents matricial calculus and determinants in a commutative ring as appliction of ``remarquable identities'' in the ring of polynomials with integer coefficients with…

We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type $\mathrm{E}_7$, and we realize groups of type…

In this paper, we consider covers of finite groups by centralizers of elements. We show that the set of centralizers that are maximal under the partial ordering form a cover of the group. We also show that the set of centralizers that are…

We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation…

The ring of Witt vectors $\mathbb{W} R$ over a base ring $R$ is an important tool in algebraic number theory and lies at the foundations of modern $p$-adic Hodge theory. $\mathbb{W} R$ has the interesting property that it constructs a ring…

We prove the centrality of $\mathrm{K}_2 (\mathsf{F}_4, \,R)$ for an arbitrary commutative ring $R$. This completes the proof of the centrality of $\mathrm K_2(\Phi,\, R)$ for any root system $\Phi$ of rank $\geq 3$. Our proof uses only…

We define the notion of an almost polynomial identity of an associative algebra $R$, and show that its existence implies the existence of an actual polynomial identity of $R$. A similar result is also obtained for Lie algebras and Jordan…

Recall that a ring R is called strongly pi-regular if, for every a in R, there is a positive integer n, depending on a, such that a^n belongs to the intersection of a^{n+1}R and Ra^{n+1}. In this paper we give a further study of the notion…

By addressing a long-standing open problem, listed in a highly regarded collection of open questions in the field and described as a "worthwhile research project", this note extends Markov's theorem (Markoff, Math. Ann., 27:177-182, 1886)…

Let T be a unital triangular algebra, let n > 1 be an integer, let gamma be an invertible element of Z(T), the center of T, and let Psi, Omega:\mathcal{T}\rightarrow \mathcal{T}$ be additive mappings satisfying \begin{align*} \Psi(X^n) =…

Let R be a strongly Z-graded ring with degree-0 subring S, and let C be a chain complex of modules over the subring P of elements of non-negative degree. We show that there are non-commutative localisations of P which detect whether the…

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is…

Let $D$ be division algebra over its center $C$, let $\sigma$ be an endormorphism of $D$, let $\delta$ be a left $\sigma$-derivation of $D$, and let $R=D[t;\sigma,\delta]$ be a skew polynomial ring. We study the structure of a class of…

The goal of this paper is to give a numerical criterion for an open question in $p$-adic Fourier theory. Let $F$ be a finite extension of $\mathbf{Q}_p$. Schneider and Teitelbaum defined and studied the character variety $\mathfrak{X}$,…

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of…

A finite group G with center Z is of central type if there exists a fully ramified character $\lambda\in\mathrm{Irr}(Z)$, i.e. the induced character $\lambda^G$ is a multiple of an irreducible character. Howlett-Isaacs have shown that G is…

In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$…

In this paper we study fundamental model-theoretic questions for free associative algebras, namely, first-order classification, decidability of the first-order theory, and definability of the set of free bases. We show that two free…