Radical Extensions of Rings
نویسنده
چکیده
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, where 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 WedderburnJacobson-Kaplansky theorems on division rings.
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