Skew derivations whose invariants satisfy a polynomial identity
Let \(R\) be an algebra over the field \(K\), \(\delta\) an additive map on \(R\), and \(\sigma\) an automorphism of \(R\). Call \(\delta\) a \(\sigma\)-derivation if \(\delta(xy)=\delta(x)y+\sigma(x)\delta(y)\) for \(x,y\in R\), and a \(q\)-skew \(\sigma\)-derivation if in addition \(\delta\sigma=q\sigma\delta\) for a nonzero \(q\in K\). The invariants of \(\delta\) are \(\{r\in R\mid\delta(r)=0\}=R^{(\delta)}\) and of \(\sigma\) are \(\{r\in R\mid\sigma(r)=r\}=R^{(\sigma)}\). Also, \(\delta\) is separable if \(R^{(\delta)}=\{r\in R\mid\delta^n(r)=0\), some \(n\geq 1\}\) and \(\sigma\) is separable if \(R^{(\sigma)}=\{r\in R\mid(\sigma-1)^n(r)=0\), some \(n\geq 1\}\). The authors investigate when the ring \(R^{(\delta)}\) satisfies a polynomial identity (PI) forces \(R\) to satisfy a PI also. The main result shows this to be the case when \(R\) is a prime algebra, \(\delta\) is \(q\)-skew \(\sigma\)-derivation, and both \(\delta\) and \(\sigma\) are algebraic over \(K\). One need only assume that \(\delta\) is a \(\sigma\)-derivation and that \(\delta\) is algebraic when \(R\) is a domain, or when \(\sigma\) is \(X\)-inner. When \(\delta\) is a separable \(q\)-skew \(\sigma\)-derivation of an arbitrary \(K\)-algebra \(R\), then \(\delta\) and \(\sigma\) algebraic force \(R\) to be a PI ring when \(R^{(\delta)}\) is, or when \(\sigma\) is separable and \(R^{(\delta)}\cap R^{(\sigma)}\) is PI.
scientific article; zbMATH DE number 1471838
scientific article