Narrow Results

In the this paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major cases, additively periodic rings remain periodic too. This includes, for instance, algebraic algebras, group rings, and matrix rings over commutative rings. Moreover, we also obtain some independent results for the new class of rings. For example, the triangular matrix rings retain that property.

Let $FG$ be the group algebra of a residually torsion free nilpotent group $G$ over a field $F$ of characteristic $0$. If $(x,y)$ is any pair of noncommuting elements of $G$, we show that for any rational number $r$ with $r\ne 0,\pm 1$, the subgroup $〈1+rx,1+ry,1+rxy〉$ is free of rank $3$ in the Malcev–Neumann field of fractions of $FG$. This result comes from a method of producing free groups of rank three in rational quaternions. In general, if a division ring $D$ has dimension $d^2$ over its center $Z$, and if the transcendence degree of $Z$ over its prime field ${𝔽}_p$ is $\ge d^2+3$, we present a method to construct free subgroups of rank three.

This paper characterizes the regularity of arbitrary near-vector spaces by representing them as direct sums of vector spaces over division rings formed by distributive elements. This approach allows us to define the dimension of a near-vector space in terms of the dimensions of these summands, providing new insights into the structure and properties of near-vector spaces.

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception.

The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

We study free subgroups in the multiplicative group of division rings generated by group rings of some classes of soluble groups.

In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decompositions into products of images of non-commutative polynomials.

Let $D$ be the field of fractions of the group algebra $k\mathrm{\Gamma }$ of the Heisenberg group $\mathrm{\Gamma }$, over the field $k$ of characteristic $\ne 2$. We show that for some involutions of $D$ that are not induced from involutions of $\mathrm{\Gamma }$, $D$ contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.

We give sufficient conditions for the existence of noncommutative free group algebras in division rings generated by group graded rings. We also relate our conclusions to already existing results on the subject improving some of them.

Let $D$ be a division ring with center $k$, $\mathrm{\text{char}}k=p\ne 2$, let ${}^{*}$ be an involution of $D$, and let $D^{†}$ be the multiplicative group of $D$. A pair $(u,v)$ is called free symmetric, if it is formed by symmetric elements, and it generates a free non-cyclic subgroup of $D^{†}$. If $U(L)$ is the enveloping algebra of the non-abelian nilpotent Lie $k$-algebra $L$ over the field $k$ of characteristic $\ne 2$, and ${}^{*}$ is a $k$-involution of $L$ extended to the field of fractions $D$ of $U(L)$, we show that $D^{†}$ contains free symmetric pairs. We also discuss the consequences of symmetric elements of a normal subgroup being torsion over the center.

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring $D[x,\sigma ,\delta ]$. We extend results of the first author and Vishkautsan on polynomial dynamics in $D[x]$. In particular, we show that if $a\in D$ and $f\in D[x,\sigma ,\delta ]$ satisfy $f(a)=a$, then $f^{\circ n}(a)=a$ for every formal power of $f$. More generally, we give a sufficient condition for a point $a$ to be $r$-periodic with respect to a polynomial $f$. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.

Let $D$ be a division ring that is algebraic over its center $F$. The aim of this paper is to describe certain classes of triangular matrices over $D$ that are similar to generalized Jordan forms, namely: (1) the class of finite triangular matrices, and (2) the class of infinite triangular matrices whose all diagonal entries, except possibly finitely many, belong to $F$.

In this paper, we demonstrate that the realm of near-vector spaces enables us to address nonlinear problems while also providing access to most of the tools that linear algebra offers. We establish fundamental results for near-vector spaces, which serve to extend classical linear algebra into the realm of near-linear algebra. Within this paper, we finalize the algebraic proof that for a given scalar group $F$, any nonempty $F$-subspace that remains stable under addition and scalar multiplication constitutes an $F$-subspace. We prove that any quotient of a near-vector space by an $F$-subspace is itself a near-vector space, along with presenting the First Isomorphism Theorem for near-vector spaces. In doing so, we obtain comprehensive descriptions of the span. By defining linear independence outside the quasi-kernel, we introduce a new concept of basis. We also establish that near-vector spaces are characterized based on the presence of a scalar basis.