JOURNAL OF ALGEBRA 123, 457477 (1989) The Zelmanov Approach to Jordan Homomorphisms of Associative Algebras KEVIN MCCRIMMON’ Department of Mathematics, University of Virginia, Math-Astronomy Building, Charlottesville, Virginia 22903-3199 Communicated by Nathan Jacobson Received June 30, 1987 Zelmanov’s work on prime Jordan algebras leads to an idempotent-free version of Martindale’s Theorem on the extension of Jordan homomorphisms and derivations from the hermitian elements H(R, *) of an associative algebra of degree >2 with involution to associative homomorphisms and derivations on R. The condition that J= H(R, *) be of degree ~2 is replaced by the intrinsic condition that J= Z(J) c H(A, *) consist entirely of values of Zelmanov polynomials. The study of Jordan homomorphisms of associative algebras has a long history. In 1942 G. Ancochea [ 1] was led in studying automorphisms of the projective line (preserving cross-ratio) to bijective additive maps D ---f D of associative division rings which preserved the product xy + yx, f(x+Y)=f(x)+f(Y)9 f(xY+Yx)= f(x)f(Y)+f(Y)f(x). He proved that if D was a quaternion algebra of characteristic 22 then such an f was an automorphism or anti-automorphism of D. In 1947 [2] he extended this to finite-dimensional simple algebras of characteristic # 2. To include characteristic 2, I. Kaplansky suggested in 1947 [lo] requiring the map f to preserve 1 and the product xyx (this being equivalent to the previous condition when l/2 is available). It has since become standard to call a linear map A -+ f B of associative @-algebras a Jordan homomorphism if it preserves the products xyx and x2 (or 1, in the case of unital algebras): (1) Ax+ Y)=f(x)+f(Y), .f(-I= d-(x) (a E @I; (11) f(xYx)=S(x)f(Y).f(x); (III) l-(x’) =f(x12 (orf(l)= 1). (0.1) * Research partially supported by NSF Grant MCS-80.02319. The author thanks the Technische Universitat Miinchen for its hospitality during the preparation of this paper. 457 0021-8693/89 $3.00 Copyright J’ 1989 by Academic Press. Inc. All rqhts ol reproductmn in any lorm rcscrvcd.