Related papers: On elements in algebras having finite number of co…

Let $F$ be a field with at least three elements and $G$ a locally finite group. This paper aims to show that if either $F$ is algebraically closed or the characteristic of $F$ is positive, then an element in the group algebra $FG$ is a…

Let R be a commutative ring. A not necessarily commutative R-algebra A is called futile if it has only finitely many R-subalgebras. In this article we relate the notion of futility to familiar properties of rings and modules. We do this by…

Let $\mathbb{F}$ denote a field with ${\rm char\,}\mathbb{F}\not=2$. The Racah algebra $\Re$ is the unital associative $\mathbb{F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$, $D$. The…

Let $\mathbb{F}$ denote a field with ${\rm char\,}\mathbb{F}\not=2$. The Racah algebra $\mathfrak{R}$ is the unital associative $\mathbb{F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$,…

A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a…

A ring $R$ with center $C$ is said to be \textit{centrally essential} if the module $R_C$ is an essential extension of the module $C_C$. In the paper, we study groups whose group algebras over fields are centrally essential rings. We focus…

We introduce the fundamental group $\r{F}(\c{A})$ of a unital $C^*$-algebra $\c{A}$ with finite dimensional trace space. The elements of fundamental group are restricted by K-theoretical obstruction and positivity. Moreover we show there…

This paper introduces and studies a new class of rings called {\it $U\sqrt{\Delta}$-rings}. A ring $R$ is $U\sqrt{\Delta}$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{\Delta(R)}$, where…

Let $\bfG$ be a connected reductive algebraic group defined over $\F_q$, where $q$ is a power of a prime $p$ that is good for $\bfG$. Let $F$ be the Frobenius morphism associated with the $\FF_q$-structure on $\bfG$ and set $G = \bfG^F$,…

It is proved that if $D$ is a $UFD$ and $R$ is a $D$-algebra, such that $U(R)\cap D\neq U(D)$, then $R$ has a maximal subring. In particular, if $R$ is a ring which either contains a unit $x$ which is not algebraic over the prime subring of…

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and $M$ be a $\mathfrak{g}$-module. The Fernando-Kac subalgebra of $\mathfrak{g}$ associated to $M$ is the subset $\mathfrak{g}[M]\subset\mathfrak{g}$ of all elements $g\in\mathfrak{g}$…

We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly…

The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

A domain $R$ is said to have the finite factorization property if every nonzero non-unit element of $R$ has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by central units. Let $k$…

A C*-algebra is n-homogeneous (where n is finite) if every its nonzero irreducible representation acts on an n-dimensional Hilbert space. An elementary proof of Fell's characterization of n-homogeneous C*-algebras (by means of their…

We investigate the structure of an alternative finite dimensional $\Q$-algebra $\mathfrak{A}$ subject to the condition that for a $\Z$-order $\Gamma \subset \mathfrak{A}$, and thus for every $\Z$-order of $\mathfrak{A}$, the loop of units…

Let $G$ be a group. A ring $R$ is called a graded ring (or $G$-graded ring) if there exist additive subgroups $R_{\alpha }$ of $R$ indexed by the elements $\alpha \in G$ such that $R=\bigoplus_{\alpha \in G}R_{\alpha }$ and $R_{\alpha…

The coprime commutators $\gamma_j^*$ and $\delta_j^*$ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. They are defined as follows. Let $G$…

Let Gamma be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H, such that if L is any…