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Then T"'(x) = y is in U since U is invariant under T. Hence Tm[x, T(x)] — [Tm(x), Tm+\x)\ = [y, T(y)] is in Z because T is centralizing on U. Since T is an automorphism, [x, T(x)] is in Z for every x in K. Apply the result of Centralizing automorphisms of prime rings [Canad. Math. Bull. 19 (1976), 113-115] to A". In the Corollary the same problem may occur and can be remedied in the automorphism case by a similar argument using the fact that the Jordan ideal is invariant under T. For derivations such that the associative ideal / is not invariant, use K = I + D(I) + D2(I) + •••, which is a nonzero ideal contained in U and invariant under D.

Erratum to: “Ideals and centralizing mappings in prime rings” [Proc. Amer. Math. Soc. 86 (1982), no. 2, 211–212; MR 83k:16025]