\(q\)-skew derivations and polynomial identities.
Let \(R\) be a prime ring and denote by \(Q\) its symmetric Martindale quotient ring. If \(\sigma\) is an automorphism of \(Q\) then \(\sigma\) is X-inner if \(\sigma\colon x\mapsto vxv^{-1}\) for some fixed unit \(v\in Q\). Otherwise \(\sigma\) is X-outer. The authors define a continuous \(\sigma\)-derivation of \(R\) as a bi-continuous automorphism of \(R\). Here bi-continuous automorphism of \(R\) means an automorphism \(\tau\) of \(Q\) such that \(I^\tau\cup I^{\tau^{-1}}\subset R\) for some nonzero ideal \(I\) of \(R\). Further a continuous \(\tau\)-derivation of \(R\) is a \(\tau\)-derivation \(\delta\) of \(Q\) such that \(I^\delta\cup I^\tau\cup I^{\tau^{-1}}\subset R\). Moreover the authors introduce the concept of a \(q\)-skew \(\sigma\)-derivation of \(R\) in the following way. Let \(R_M\) be the (left) Martindale ring of quotients of \(R\), and \(C\) the centre of \(R_M\) (that is, the extended centroid of \(R\)). Now if \(\delta\) is a continuous \(\sigma\)-derivation of \(R\) and \(S\subset R_M\) set \(S^{(\delta)}\) to be \(\{r\in S\mid r^\delta=0\}\), and \(S^{(\sigma)}=\{r\in S\mid r^\sigma=r\}\). Then \(\delta\) is \(q\)-skew if \(\sigma\delta\sigma^{-1}=q\delta\) for some \(q\in C^{(\sigma)}\cap C^{(\delta)}\). The main result of the paper under review treats continuous \(q\)-skew \(\sigma\)-derivations \(\delta\) of the prime ring \(R\). If in addition \(\delta\) is left \(R_M\)-algebraic and if \(R^{(\delta)}\) satisfies a nontrivial polynomial identity then \(R\) is PI as well.
scientific article; zbMATH DE number 2156088