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

DOI:
https://doi.org/10.1090/S0002-9939-1984-0728345-7

MathSciNet review:
728345

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a ring satisfying the maximal condition on annihilator left ideals and be an automorphism of . We show that the Jacobson radical of the skew polynomial ring is nonzero if and only if the prime radical of is nonzero. Furthermore, it is so if and only if the prime radical of is nonzero. In general, an example is given of a commutative semisimple algebra and an automorphism such that is prime but the Levitzki radical of is nonzero.

**[1]**S. S. Bedi and J. Ram,*Jacobson radical of skew polynomial rings and group rings*, Israel J. Math.**35**(1980), 327-338. MR**594338 (82d:16006)****[2]**A. Goldie and G. Michler,*Ore extensions and polycyclic group rings*, J. London Math. Soc.**9**(1974-75), 337-345. MR**0357500 (50:9968)****[3]**R. L. Graham and B. L. Rothschild,*A short prof of Van der Waerden's theorem on arithmetic progressions*, Proc. Amer. Math. Soc.**42**(1974), 385-386. MR**0329917 (48:8257)****[4]**C. R. Jordan and D. A. Jordan,*A note on semiprimitivity of ore extensions*, Comm. Algebra**4**(1976), 647-656. MR**0404314 (53:8116)****[5]**M. B. Nathanson,*Arithmetic progressions contained in sequences with bounded gaps*, Canad. Math. Bull.**23**(1980), 491-493. MR**602607 (82c:10067)****[6]**D. S. Passman,*Algebraic structure of group rings*, Interscience, New York, 1977. MR**470211 (81d:16001)****[7]**B. L. Van der Waerden,*Beweis einer Baudet'schen Vermutung*, Nieuw Arch. Wisk.**15**(1927), 212-216.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
16A05,
16A12,
16A20

Retrieve articles in all journals with MSC: 16A05, 16A12, 16A20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1984-0728345-7

Article copyright:
© Copyright 1984
American Mathematical Society