The rank condition and strong rank conditions for Ore extensions

Let $R$ be a ring, $\sigma :R\to R$ a ring endomorphism, and $\delta :R\to R$ a $\sigma$-derivation. We establish that the Ore extension $R[x;\sigma ,\delta ]$ satisfies the rank condition if and only if $R$ does. In addition, we prove analogous results for the right and left strong rank conditions. However, in the right case, the if" part requires the hypothesis that $σ$ is an automorphism, whereas, in the left case, this assumption is needed for the only if" part. Finally, we provide a new proof of an old result of Susan Montgomery stating that a skew power series ring is directly (respectively, stably) finite if and only if its coefficient ring is directly (respectively, stably) finite.