Lie ideals and differentiations of semiprime rings (Q913925)

The author announces some results, without proof, most of which extend theorems of \textit{E. C. Posner} [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)] to semi-prime rings. Let \(R\) be a semi-prime ring and let \(U\) be a Lie ideal of \(R\) whose intersection with each nonzero ideal of \(R\) is not central. The first result claims that if \(R\) is 2-torsion free and if \(U\) satisfies a polynomial identity of degree \(n\), then \(R\) satisfies a polynomial identity of degree \(2n\). The other results involve the orthogonal completion \(K(R)\) of \(R\). Two statements are that if \(R\) is 2- torsion free and if \(d_ 1,d_ 2\in\text{Der}(R)\) so that either \(d_ 1d_ 2\in\text{Der}(R)\) or \(d_ 1d_ 2(U)=0\), then \(K(R)=R_ 1\oplus R_ 2\) where each \(R_ i\) is invariant under \(d_ j\) and \(d_ i(R_ i)=0\). Finally, if \(d\in\text{Der}(R)\) and \([d(r),r]\) is central for all \(r\in R\), then \(K(R)=R_ 1\oplus R_ 2\) with \(d(R_ 1)=0\), \(d(R_ 2)\subset R_ 2\), and \(R_ 2\) is commutative.