Generalized algebraic rational identities of subnormal subgroups in division rings

TL;DR

This paper introduces a new concept called generalized algebraic rational identities to analyze division rings, showing that certain identities imply the division ring is finite-dimensional over its center.

Contribution

It defines generalized algebraic rational identities and proves that their presence in subnormal subgroups constrains the division ring to be finite-dimensional.

Findings

Subnormal subgroups satisfying non-trivial identities imply finite dimensionality.

Generalizes previous results on identities in division rings.

Provides a new tool for studying the structure of division rings.

Abstract

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 $D^*$ of a division ring $D$ with center $F$ satisfies a non-trivial generalized algebraic rational identity of bounded degree, then $D$ is a finite dimensional vector space over $F$. This generalizes some previous results.