Invariants of skew derivations

Abstract:

If $\sigma$ is an automorphism and $\delta$ is a $\sigma$-derivation of a ring $R$, then the subring of invariants is the set $R^{(\delta )} = \{r \in R \mid \delta (r) = 0 \}.$ The main result of this paper is

Theorem. Let $\delta$ be a $\sigma$-derivation of an algebra $R$ over a commutative ring $K$ such that \begin{equation*}\delta ^{n+k}(r) + a_{n-1}\delta ^{n+k-1}(r) + \dots + a_{1}\delta ^{k+1}(r) + a_{0}\delta ^{k}(r) =0, \end{equation*} for all $r \in R$, where $a_{n-1}, \dots , a_{1},a_{0} \in K$ and ${a_{0}}^{-1} \in K$.

If $R^{n+1} \not = 0$, then $R^{(\delta )} \not = 0$.

If $L$ is a $\delta$-stable left ideal of $R$ such that $l.ann_{R}(L) = 0$, then $L^{(\delta )} \not = 0$.

This theorem generalizes results on the invariants of automorphisms and derivations.