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arXiv · 2505.21030

The rank condition and strong rank conditions for Ore extensions

Karl Lorensen·Johan Öinert
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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.

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Karl Lorensen, Johan Öinert. 2025-05-27. The rank condition and strong rank conditions for Ore extensions. https://doi.org/10.1142/s0219498827501738

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