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Ci, cij C I . This is a subset of of m Let I L be the maximal locally be a locally nilpotent left ideal. We claim this is locally nilpotent. I + IA and write b i = ci + Let ~ cijaij . 13 I so there exists an of these elements is O. m Any product of such that any product r of the bi is 32 a sum of terms each a product of duct multiplied duct of m r elements of on the right by an element of of these is O and I + IA The maximal nil (locally nil) radical (Levitzki n i l radical) the upper nil radical of S A.

Hence f is non-degenerate on if its extension is non-degenerate on field of the base field. bilinear form Mn(K). is a base for a If tr(a,b) a = ~ aijeij tr(a'ekg ) = 0 for all VF for F V if and only an extension Hence it suffices to prove that the trace is non-degenerate on a matrix algebra one has k, 6 tr(a,e k £ ) implies = a 6k" a = O. Hence Thus tr(, ) is non-degenerate. The foregoing results are applicable to primitive algebras satisfying proper identities. If n is the degree of such an algebra then the Formanek polynomial for polynomial for A.

THEOREM 3. K. [ a] ] = is central sJ~nple. a field F = K Then is central simple over a field A~B DEFINITION. K -~K = 0 J is a splitting field. F is To see the second statement we refer to the proof of the Kaplansky-Amitsur theorem. There we showed that if transformations in subfield of A V/~, A then transformations in F is a maximal is a dense algebra of linear This applies in particular to dimensional central simple. finite dimensional over is a dense algebra of linear a division algebra and A' = FLA V/F.

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