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On classical quotients of polynomial identity rings with involution


Author: Louis Halle Rowen
Journal: Proc. Amer. Math. Soc. 40 (1973), 23-29
MSC: Primary 16A28
DOI: https://doi.org/10.1090/S0002-9939-1973-0323822-8
MathSciNet review: 0323822
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Abstract: Let $ (R, \ast )$ denote a ring $ R$ with involution $ ( \ast )$, where ``involution'' means `` $ {\text{anti - automorphism of order}} \leqq {\text{ two}}$". We can specialize many ring-theoretical concepts to rings with involution; in particular an ideal of $ (R, \ast )$ is an ideal of $ R$ stable under $ ( \ast )$, and the center of $ (R, \ast )$ is the set of central elements of $ R$ which are fixed under $ ( \ast )$. Then we say $ (R, \ast )$ is prime when the product of any two nonzero ideals of $ (R, \ast )$ is nonzero; similarly $ (R, \ast )$ is semiprime when any power of a nonzero ideal of $ (R, \ast )$ is nonzero. The main result of this paper is a strong analogue to Posner's theorem [5], namely that any prime $ (R, \ast )$ with polynomial identity has a ring of quotients $ {R_T}$, formed merely by adjoining inverses of nonzero elements of the center of $ (R, \ast )$. This quotient ring $ ({R_T}, \ast )$ is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.


References [Enhancements On Off] (What's this?)

  • [1] S. A. Amitsur, Prime rings having polynomial identities, Proc. London Math. Soc. (3) 17 (1967), 470-486. MR 36 #209. MR 0217118 (36:209)
  • [2] -, Identities in rings with involutions, Israel J. Math. 7 (1969), 63-68. MR 39 #4216. MR 0242889 (39:4216)
  • [3] N. Jacobson, Structure of rings, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R.I., 1964. Chapter X and Appendix B. MR 36 #5158. MR 0222106 (36:5158)
  • [4] -, Structure and representation of Jordan algebras, Amer. Math. Soc. Colloq. Publ., vol. 39, Amer. Math. Soc., Providence, R.I., 1968. Sections I.4 and V.7. MR 40 #4330. MR 0251099 (40:4330)
  • [5] W. S. Martindale III, Rings with involution and polynomial identities, J. Algebra 11 (1969), 186-194. MR 38 #3302. MR 0234990 (38:3302)
  • [6] L. H. Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219-223. MR 0309996 (46:9099)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0323822-8
Keywords: Classical ring of quotients, Goldie ring, involution, polynomial identity, prime, semiprime, simple, center, central quotients
Article copyright: © Copyright 1973 American Mathematical Society

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