The rank condition and strong rank conditions for Ore extensions

TL;DR

This paper investigates the conditions under which Ore extensions satisfy rank conditions, establishing equivalences with the base ring and providing new proofs for related finiteness properties.

Contribution

It proves that Ore extensions satisfy the rank condition if and only if the base ring does, and offers new proofs for finiteness properties of skew power series rings.

Findings

Ore extension satisfies rank condition iff base ring does

New proof of finiteness equivalence for skew power series rings

Different conditions for left and right strong rank conditions

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.