The rank condition and strong rank conditions for Ore extensions
Karl Lorensen, Johan Öinert
TL;DR
The paper proves that the rank condition for an Ore extension $R[x;\sigma,\delta]$ holds iff it holds for the base ring $R$, and it analyzes the left/right strong rank conditions, showing automorphism of $\sigma$ is required in certain directions. It introduces a filtration method to reduce these properties to the base ring via Propositions 1.9 and 1.10, and demonstrates nuanced differences between left and right cases, including necessity results for automorphism hypotheses (via Lemma 1.11). Additionally, it provides a new proof that skew power series rings are directly (respectively, stably) finite exactly when their coefficient rings are, with broader implications for Weyl rings. The work links these ring-theoretic properties to broader questions and opens several avenues for further questions about Ore extensions.
Abstract
Let $R$ be a ring, $σ:R\to R$ a ring endomorphism, and $δ:R\to R$ 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.