Smash products and differential identities. (Q2841348)

Let \(U\) be the universal enveloping algebra of a Lie algebra and \(R\) a \(U\)-module algebra, where \(U\) is considered as a Hopf algebra canonically. The authors of the paper determine the centralizer of \(R\) in \( R\#U\) with its associated graded algebra. Further, they apply this to the Ore extension \(R[X;\varphi]\), where \(\varphi\colon X\to\mathrm{Der}(R)\). Moreover, with the help of PBW-bases, they prove the following: Let \(R\) be a prime ring and \(Q\) the symmetric Martindale quotient ring of \(R\). For \(f_i,g_i\in Q[X;\varphi]\), \(\sum_if_irg_i=0\) for all \(r\in R\) iff \(\sum_if_i\otimes g_i=0\), where \(\otimes\) is over the centralizer of \(R\) in \(Q[X;\varphi]\). At the end, the authors deduce from the above mentioned results Kharchenko's theorem on differential identities.