Related papers: On subgroups in division rings of type $2$

In this article, we identify the existence of a divisibility relationship between the number of ring homomorphisms and surjective group homomorphisms. We demonstrate that for finite cyclic structures, the number of ring homomorphisms from…

A polynomial $f(t)$ in an Ore extension $K[t;S,D]$ over a division ring $K$ is a Wedderburn polynomial if $f(t)$ is monic and is the minimal polynomial of an algebraic subset of $K$. These polynomials have been studied in "Wedderburn…

In this paper, we continue our study of the class of diagram groups. Simply speaking, a diagram is a labelled plane graph bounded by a pair of paths (the top path and the bottom path). To multiply two diagrams, one simply identifies the top…

In 2012 the second author obtained a description of the lattice of subgroupsof a Chevalley group $G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided $\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$…

Our main result states that a finite semiring of order >2 with zero which is not a ring is congruence-simple if and only if it is isomorphic to a `dense' subsemiring of the endomorphism semiring of a finite idempotent commutative monoid. We…

Let $R$ be a ring with unity and $U(R)$ its group of units. Let $\Delta U=\{a\in U(R)\mid [U(R):C_{U(R)}(a)]<\infty\}$ be the $FC$-radical of $U(R)$ and let $\nabla(R)=\{a\in R\mid [U(R):C_{U(R)}(a)]<\infty\}$ be the $FC$-subring of $R$. An…

Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively…

Let $A\subset B$ be an integral ring extension of integral domains with fields of fractions $K$ and $L$, respectively. The integral degree of $A\subset B$, denoted by ${\rm d}_A(B)$, is defined as the supremum of the degrees of minimal…

In this paper, we introduce a new class of rings whose elements are a sum of a central element and a nilpotent element, namely, a ring $R$ is called$CN$ if each element $a$ of $R$ has a decomposition $a = c + n$ where $c$ is central and $n$…

We study finite groups $G$ having a subgroup $H$ and $D \subset G \setminus H$ such that the multiset $\{ xy^{-1}:x,y \in D\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\it difference set}). We…

We prove the finiteness of the genus of finite-dimensional division algebras over many infinitely generated fields. More precisely, let $K$ be a finite field extension of a field which is a purely transcendental extension of infinite…

This paper began as an investigation of the question of whether $D_1 \otimes_F D_2$ is a domain where the $D_i$ are division algebras and $F$ is an algebraically closed field contained in their centers. We present an example where the…

Milliet asks the following question: given two prime numbers $p\neq q$, is there a division algebra of characteristic $p$ which is of dp-rank $q^2$ and of dimension $q^2$ over its center? We answer in the affirmative. We also give an…

A dimension group is a partially ordered countable group such that (1) every finite subset is contained in an ordered subgroup which is a finite direct power of Z and (2) the group has an order unit i.e. a positive element u such that every…

Let $D$ be a non-commutative division ring, $G$ a subnormal subgroup of ${\mathrm GL}_n(D)$. In this note we show that if $G$ contains a non-abelian solvable maximal subgroup, then $n=1$ and $D$ is a cyclic algebra of prime degree over $F$.

Let $ m , n \in \mathbb{N}$, $D$ be a division ring, and $M_{m \times n}(D)$ denote the bimodule of all $m \times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m \times n}(D)$ in terms of left row…

For a given free group $F$ of arbitrary rank (possibly infinite), and its subgroup $G$, we address the question whether a lower central subgroup of $G$ can contain a lower central subgroup of $F$. We show that the answer is no if $G$ does…

Given an element $f$ in a regular local ring, we study matrix factorizations of $f$ with $d \ge 2$ factors, that is, we study tuples of square matrices $(\varphi_1,\varphi_2,\dots,\varphi_d)$ such that their product is $f$ times an identity…

A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their…

Let $D$ be a division ring with infinite center $F$; $\sigma$ be an anti-automorphism of $D$ and $m$ be a positive integer such that $\sigma^m\neq \mathrm{Id}$. In this paper, we show that if $D$ satisfies a $\sigma^m$-GRI, then $D$ is…