Prime nondegenerate Jordan algebras with nonzero socle and the symmetric Martindale algebra of quotients (Q805727)

The author studies in detail the symmetric Martindale algebra \(Q_ s(A)\) of quotients of a prime associative algebra A with nonzero socle. He shows the existence of a pairing of vector spaces X and Y such that \(Q_ s(A)\) is identified with the automorphisms of X having an adjoint. Combining this with Zelmanov's classification of prime nondegenerate Jordan algebras [\textit{E. I. Zel'manov}, Sib. Math. J. 24, 73-85 (1983); translation from Sib. Mat. Zh. 24, No.1(137), 89-104 (1983; Zbl 0534.17009)] the author rederives a structure theorem of Osborn and Racine for prime Jordan algebras.