Invariant polynomials of Ore extensions by $q$-skew derivations

Abstract:

Let $R$ be a prime ring with the symmetric Martindale quotient ring $Q$. Suppose that $\delta$ is a quasi-algebraic $q$-skew $\sigma$-derivation of $R$. For a minimal monic semi-invariant polynomial $\pi (t)$ of $Q[t;\sigma ,\delta ]$, we show that $\pi (t)$ is also invariant if $\textrm {char} R=0$ and that either $\pi (t)-c$ for some $c\in Q$ or $\pi (t)^p$ is a minimal monic invariant polynomial if $\textrm {char} R=p\ge 2$. As an application, we prove that any $R$-disjoint prime ideal of $R[t;\sigma ,\delta ]$ is the principal ideal $\langle p(t)\rangle$ for an irreducible monic invariant polynomial $p(t)$ unless $\sigma$ or $\delta$ is X-inner.