On zero-divisors in group rings of groups with torsion

TL;DR

This paper investigates the existence of nontrivial zero-divisors in group rings, proving their presence in certain complex group structures like free Burnside groups and free products with torsion.

Contribution

It establishes the existence of nontrivial zero-divisors in group rings of free Burnside groups of large odd exponent and free products with torsion, advancing understanding in algebraic structures.

Findings

Nontrivial zero-divisors exist in group rings of free Burnside groups with large odd exponents.

Such zero-divisors are also present in group rings of free products of groups with torsion.

The paper solves an open problem regarding zero-divisors in specific group rings.

Abstract

Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.