A Theorem on Rings

Abstract

In a recent paper, Kaplansky [2] proved the following theorem: Let R be a ring with centre Z, and such that xn(x) ∈ Z for every x∈ R. If R, in addition, is semi-simple then it is also commutativeThe existence of non-commutative rings in which every element is nilpotent rules out the possibility of extending this result to all rings. One might hope, however, that if R is such that xn(x) ∈ Z for all x ∈ R and the nilpotent elements of R are reasonably “well-behaved,” then Kaplansky's theorem should be true without the restriction of semi-simplicity.