Locally nilpotent skew extensions of rings

2020/01/12 by Piotr Grzeszczuk, Grzeszczuk, Piotr
Mathematics · #16N20 #16N40 #16W25 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16N20 #msc:16N40 #msc:16W25

paper · pdf · doi:10.48550/arxiv.2001.03881

final version, to appear in Journal of Pure and Applied Algebra

arxiv created 2020/03/04 · arxiv updated 2020/03/05

Abstract

We extend existing results on locally nilpotent differential polynomial rings to skew extensions of rings. We prove that if \mathscrG=\σ_t\_{t∈ T} is a locally finite family of automorphisms of an algebra R, \mathscrD=\δ_t\_{t∈ T} is a family of skew derivations of R such that the prime radical P of R is strongly invariant under \mathscrD, then the ideal P⟨ T,\mathscrG,\mathscrD⟩^* of R⟨ T,\mathscrG,\mathscrD⟩, generated by P, is locally nilpotent. We then apply this result to algebras with locally nilpotent derivations. We prove that any algebra R over a field of characteristic 0, having a surjective locally nilpotent derivation d with commutative kernel, and such that R is generated by ker d², has a locally nilpotent Jacobson radical.

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