Let R be a ring, σ : R → R a ring endomorphism, and δ a σ-derivation. We establish that the Ore extension R[x; σ, δ] 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.