The rank condition and strong rank conditions for Ore extensions

Let [Formula: see text] be a ring, [Formula: see text] a ring endomorphism, and [Formula: see text] a [Formula: see text]-derivation. We establish that the Ore extension [Formula: see text] satisfies the rank condition if and only if [Formula: see text] 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 [Formula: see text] 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.