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 − y∗x∗) = 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.
