Related papers: On normal subgroups of division rings which are ra…

Let $D$ be a division ring with the center $F=Z(D)$. Suppose that $N$ is a normal subgroup of $D^*$ which is radical over $F$, that is, for any element $x\in N$, there exists a positive integer $n_x$, such that $x^{n_x}\in F$. In…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

We develop the theory of group graded division ring parallel to the one by P. Cohn for (ungraded) division rings.

We generalize the idea of a Schur ring of a group to the category of semigroups. Fundamental results of Schur rings over groups are shown to be true for Schur rings over semigroups. Examples where Schur rings differ between the two…

In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of…

We prove a division algorithm for group rings of high genus surface groups and use it to show that some $2$-complexes with surface fundamental groups are standard. We also give an application of division to cohomological dimension of…

Finite groups that are embeddable in the multiplicative groups of division rings $K$ were completely determined by S. A. Amitsur in 1955. In case $K$ has characteristic $p>0$, the only possible finite subgroups of $K^*$ are cyclic groups,…

In \cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In a…

The work of Chatzidakis and Hrushovski on the model theory of difference fields in characteristic zero showed that groups defined by difference equations have a very restricted structure. Recent work of Chatzidakis, Hrushovski and Peterzil…

Let $D$ be a division ring with infinite center, $K$ a proper division subring of $D$ and $N$ an almost subnormal subgroup of the multiplicative group $D^*$ of $D$. The aim of this paper is to show that if $K$ is $N$-invariant and $N$ is…

We investigate small $p$-groups with cohomology rings of depth higher than predicted by Duflot's theorem. In these groups, a sampling would suggest several naive conjectures about the degrees of the additional regular sequence elements. We…

In this note, we introduce a new concept of a {\it generalized algebraic rational identity} to investigate the structure of division rings. The main theorem asserts that if a non-central subnormal subgroup $N$ of the multiplicative group…

Invertibility is important in ring theory because it enables division and facilitates solving equations. Moreover, (nonassociative) rings can be endowed with an extra ''structure'' such as order and topology allowing more richness in the…

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely…

Let R[G] be the group ring of a group G over an associative ring R with unity such that all prime divisors of orders of elements of G are invertible in R. If R is finite and G is a Chernikov (torsion FC-) group, then each R-derivation of…

The notion of derivation with invertible values as a derivation of ring with unity that only takes multiplicatively invertible or zero values appeared in a paper of Bergen, Herstein and Lanski, in which they determined the structure of…

In the second section, we introduce hemiring-valued pseudonormed rings and generalize Albert's result which states that every finite-dimensional algebra can be normed. Next, we introduce shrinkable hemirings and prove that dense division…

We provide new characterizations of pseudo-Frobenius and quasi-Frobenius rings in terms of tight modules. In the process, we also provide fresh perspectives on FGF and CF conjectures. In particular, we propose new natural extensions of…

Consideration of certain properties of group rings and their ideals search

In this paper, we prove an extension of Zaks' conjecture on integral domains with semi-regular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zero-divisors). The main result…