On subgroups in division rings of type $2$

TL;DR

This paper investigates the structure of multiplicative subgroups within division rings of type 2, where each generated subring over the center is finite-dimensional, providing insights into their algebraic properties.

Contribution

It introduces the concept of division rings of type 2 and analyzes the properties of their multiplicative subgroups, a novel focus in division ring theory.

Findings

Characterization of subgroups in division rings of type 2

Conditions for subgroups to be finite or infinite

Structural properties of subgroups in these rings

Abstract

Let $D$ be a division ring with center $F$. We say that $D$ is a {\em division ring of type $2$} if for every two elements $x, y\in D,$ the division subring $F(x, y)$ is a finite dimensional vector space over $F$. In this paper we investigate multiplicative subgroups in such a ring.