A condition for commutativity in a domain
Volume 179 / 2025
Colloquium Mathematicum 179 (2025), 45-53
MSC: Primary 16U10
DOI: 10.4064/cm9592-8-2025
Published online: 21 September 2025
Abstract
Let $R$ be a unitary ring without zero divisors. We prove that if $R$ contains an element with finite centralizer, then $R$ must be commutative. Furthermore, employing Zorn’s Lemma (which is equivalent to the Axiom of Choice), we demonstrate that in the noncommutative case, every element of $R$ is contained in an infinite commutative subring.
