Portal de Periódicos da CAPES

Lie isomorphisms of primitive rings

1963; American Mathematical Society; Volume: 14; Issue: 6 Linguagem: Inglês

10.1090/s0002-9939-1963-0160798-4

1088-6826

Wallace S. Martindale,

Algebraic Geometry and Number Theory

for all x, yER. In this paper we study Lie isomorphisms of a primitive ring R onto a primitive ring R', where we assume that the characteristic of R is different from 2 and 3 and that R contains three nonzero orthogonal idempotents whose sum is the identity. Such isomorphisms will be shown to be of the form o+r, where ois either an isomorphism or the negative of an anti-isomorphism of R into a primitive ring L' containing R' and r is an additive mapping of R into the center of L' which maps commutators into zero. This generalizes a theorem of Hua [1], who obtained the above result in the case that R (= '= L') was the ring of all n X n matrices over a division ring, n> 2. On the other hand, due to our requirement concerning idempotents, we fall far short of providing a general solution to Herstein's conjecture [2] that the result holds for arbitrary simple rings. An important part of our proof consists of a repetition of arguments involving matrix units used by Hua [1], and for the sake of completeness (and also because of the relative inaccessability of Hua's paper) we shall reproduce his proofs in some detail when the occasion demands. The author is especially grateful to Professor Nathan Jacobson for several valuable comments towards improving an earlier version of this paper. In the case of simple rings he indicated how all calculations involving matrix units can be eliminated, and he suggested extending our original result to the more general case of primitive rings.