The rank condition and strong rank conditions for Ore extensions

Abstract

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 $\sigma$ 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.

Final version. To appear in Journal of Algebra and its Applications