Vol. 43, No. 1, 1972

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Strong Lie ideals

Albert Joseph Karam

Vol. 43 (1972), No. 1, 157–169

R is 2-torsion free semiprime with 2R = R. A Lie ideal, U, of R-strong if aua U for all a R,u U. One shows that U contains a nonzero two-sided ideal of R. If R has an involution, , (with skew-symmetric elements K) a Lie ideal, U, of K is K-strong if kuk U for all k K,u U. It is shown that if R is simple with characteristic not 2 and either the center, Z, is zero or the dimension of R over the center is greater than 4, then U = K. If R is a topological annihilator ring with continuous involution and if U is closed K-strong Lie ideal, U = C K where C is a closed two-sided ideal of R. A Lie ideal, U, of K is HK-strong if u3 U for all u U. A result similar to the above result for K-strong Lie ideals can be shown. Let R be a simple ring with involution such that Z = (0) or the dimension of R over Z is greater than 4. Let ϕ be a nonzero additive map from R into a ring A such that the subring of A generated by {ϕ(x) : x R} is a noncommutative, 2-torsion free prime ring. Suppose ϕ(xy yx) = ϕ(x)ϕ(y) ϕ(y)ϕ(x) for all x,y R. As an application of the above theory, ϕ is shown to be an associative isomorphism.

Mathematical Subject Classification
Primary: 16A68
Received: 26 July 1971
Published: 1 October 1972
Albert Joseph Karam