We show that if $\phi$ is an automorphism of $R$, then all prime ideals of the skew polynomial ring $R[x;\phi]$ are right Goldie if and only if (a) every prime ideal of $R$ is right Goldie and (b) every strongly $\phi$-prime ideal of $R$ is a finite intersection of prime ideals. [We say $I$ is strongly $\phi$-prime if $\phi(I)=I$ and if whenever $\phi^n(J)K$ is contained in $I$ for ideals $J,K$ and all large $n$, we have either $J$ or $K$ contained in $I$.] We obtain a similar result for skew Laurent rings, and we obtain some partial results for skew polynomial rings with both automorphism and derivation. We show that if $R$ has the a.c.c. on ideals, then all prime ideals in $R[x;\phi]$ are right Goldie if and only if all prime ideals in $R$ are right Goldie, but we give an example showing that this is not true in general. We give some other negative examples.
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