Generalized derivations as Jordan homomorphisms on Lie ideals and right ideals.
A generalized derivation of a ring \(R\) is an additive \(g\colon R\to R\) so that for all \(x,y\in R\), \(g(xy)=g(x)y+xd(y)\) for \(d\) a derivation of \(R\). For this paper let \(R\) be a prime ring with noncentral Lie ideal \(L\), nonzero right ideal \(T\), and nonzero generalized derivation \(g\). The author proves two main results when \(g\) acts like a Jordan homomorphism. The first assumes that for all \(u\in L\), \(g(u^2)=g(u)^2\), and proves that \(g(r)=r\) for all \(r\in R\), or else \(\text{char\,}R=2\) and \(R\) embeds in \(M_2(F)\) for \(F\) a field. The second main result replaces \(T\) for \(U\) above and concludes that either: (1) \(\text{char\,}R=2\) and \(S_4(x_1,x_2,x_3,x_4)x_5\) is an identity for \(T\), where \(S_4\) is the standard identity of degree four; (2) \([T,T]T=0\); (3) for all \(r\in R\), \(g(r)=ar\) for some \(a\in R\) so that \(aT=0\); (4) \(g(t)=t\) for all \(t\in T\); or (5) for some \(q\in R\), \(g(t)=tq\) and \(qt=t\) for all \(t\in T\).
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