2025/05/27 by Lorensen, Karl, Öinert, Johan
#16P99 #16S32 #16S36 #16S99 #16W25 #16W70 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2505.21030
Let R be a ring, σ:R→ R a ring endomorphism, and δ:R→ 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.