A condition for commutativity in a domain

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.