Jordan mappings of semiprime rings

Abstract

An additive mapping 8 of a ring R into a 2-torsion free ring R’ is called a Jordan homomorphism if 6(ab +&z)=@(a) 8(b)+ 6(b) @(a) for all a, bE R. A well-known result of I. N. Herstein [4] states that every Jordan homomorphism onto a prime ring is either a homomorphism or an antihomomorphism. Suppose that R’ contains ideals U’ and V’ with null intersection. Let cp: R -+ U’ be a homomorphism and II/: R + V’ be an antihomomorphism. A mapping 6 = 40 + + is a so-called direct sum of mappings rp and 11/. Obviously, B is a Jordan homomorphism. According to this construction we see that Herstein’s result [4] does not hold in semip~me rings. Moreover, W. E. Baxter and W. S. Martindale El ] showed by an example that a Jordan homomorphism 0 of a ring R onto a semiprime ring R’ is not necessarily a direct sum of a homomorphism and an antihomomorphism. But they proved that there always exists an essential ideal E of R such that the restriction of 8 to E is a direct sum of a homomorphism cp: E -+ R’ and an antihomomorphism y5: E + R’ [ 1, Theorem 2.71. In this paper (Theorem 2.3) we extend this result by showing that E can be choosen so that it is a sum of ideals U and V of R such that cp vanishes on V and tc/ vanishes on U. Even more, for each x E R we have B(ux) = e(u) e(x) for all u E U and B(ux) = f?(x) e(o) for all u E V. This result also contains an affirmative answer to the question of Baxter and Martindale, whether there is a way to choose the ideal E so that 8(E) is an associative subring of R’. As we shall see, in our case 6(E) is the essential (associative) ideal of R’. An additive mapping 0 of a ring R into a ring R’ which satisfies B(aba) = &a) 6(b) 6(a) for all a, b E R will be called a Jordan triple homomorphism. An easy computation shows that every Jordan homomorphism is also a