Jacobson's generalization [5, Theorem 8] of Wedderburn's theorem [8] states that an algebraic division algebra over a finite field is commutative. These algebras have the property that some power2 of each element lies in the center. Kaplansky observed in [7] that any division ring, or, more generally, any semisimple ring, in which some power of each element lies in the center is commutative. Kaplansky's idea was generalized in [l], and radical extensions of arbitrary subrings were studied,

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... the ring A is a radical extension of the ring B in case each aÇ0.A is radical over B in the sense that some power of a lies in B. In this connection Theorem A of [l ] states: If A is a simple ring with a minimal one-sided ideal, and radical over a subring B^A, then A is a field. This is the best possible result of this type for which A/B is radical, and no restriction is placed on B (best in the sense that there exist noncommutative primitive rings with minimal one-sided ideals and radical over proper simple subrings [l, §4]). The starting point for the present investigation is the observation that if A is a division ring and radical over center, then A is a radical extension of both a division subring, and a commutative subring. Accordingly, the present paper is devoted to the study of rings A which are radical over (1) division subrings, or (2) commutative subrings. In connection with (1), there exists the following generalization of the Wedderburn-Jacobson-Kaplansky theorems on division rings. Theorem 1.1. If A is a ring with no nil ideals y¿ {o}, and if A is radical over a division subring B¿¿A, then A is a field. Corresponding to (2), one has the following extension of Kaplansky's theorem for semisimple rings. In it (and throughout this paper) J(A) denotes the Jacobson radical of the ring A. Theorem 1.2. If A is radical over a commutative subring B, and if J(A) = {0}, then A is commutative. Kaplansky's theorem [7] is, then, the special case of Theorem 1.2 for which B is central.