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.
Lorensen et al. (Fri,) studied this question.