Invariant polynomials of Ore extensions by \(q\)-skew derivations. (Q2846728)

Let \(R\) be a prime ring with a symmetric Martindale quotient ring \(Q\) and \(Q[t;\sigma,\delta]\) the Ore extension of \(Q\) by a \(\sigma\)-derivation \(\delta\). \(f(t)\in Q[t;\sigma,\delta]\) is a cv-polynomial if it is associated to an automorphism \(\tau\) of \(Q\) such that \(f(t)r-\tau(r)f(t)\in Q\) for any \(r\in R\). A cv-polynomial \(f(t)\) is semi-invariant if \(f(t)r-\tau(r)f(t)=0\) for all \(r\in R\). A polynomial \(f(t)\) is invariant if it is semi-invariant and \(f(t)t=(at+b)f(t)\) for some \(a,b\in Q\).NEWLINENEWLINE Main theorem: for a minimal monic semi-invariant polynomial \(\pi(t)\in Q[t;\sigma,\delta]\), if \(\text{char\,}R=0\), \(\pi(t)\) is also invariant and if \(\text{char\,}R=p\geq 2\), then either \(\pi(t)-c\) for some \(c\in Q\) or \(\pi(t)^p\) is a minimal monic invariant polynomial. As an application of the main theorem, the authors 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.