This paper shows that finite centralizers imply commutativity in unitary rings, highlighting the importance of Zorn's Lemma.
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.
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Abderrahim Makki Naciri (2025) studied this question.
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