Extremely noncommutative elements in rings

We study rings and K-algebras in which all elements or all noncentral elements have smallest possible centralizer. Our principal result asserts that a ring R must be either finite or commutative if each noncentral element a has centralizer equal to the subring generated by a. Supported by the Natural Sciences and Engineering Research Council of Canada, Grant 3961. Authors’ addresses: Howard E. Bell, Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1; Abraham A. Klein, Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel