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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the semisimplicity of skew polynomial rings


Author: Jai Ram
Journal: Proc. Amer. Math. Soc. 90 (1984), 347-351
MSC: Primary 16A05; Secondary 16A12, 16A20
MathSciNet review: 728345
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Abstract: Let $ R$ be a ring satisfying the maximal condition on annihilator left ideals and $ \sigma $ be an automorphism of $ R$. We show that the Jacobson radical of the skew polynomial ring $ {R_\sigma }[x]$ is nonzero if and only if the prime radical of $ {R_\sigma }[x]$ is nonzero. Furthermore, it is so if and only if the prime radical of $ R$ is nonzero. In general, an example is given of a commutative semisimple algebra $ R$ and an automorphism $ \sigma $ such that $ {R_\sigma }[x]$ is prime but the Levitzki radical of $ {R_\sigma }[x]$ is nonzero.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0728345-7
PII: S 0002-9939(1984)0728345-7
Article copyright: © Copyright 1984 American Mathematical Society