On classical quotients of polynomial identity rings with involution
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- by Louis Halle Rowen
- Proc. Amer. Math. Soc. 40 (1973), 23-29
- DOI: https://doi.org/10.1090/S0002-9939-1973-0323822-8
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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
- S. A. Amitsur, Prime rings having polynomial identities with arbitrary coefficients, Proc. London Math. Soc. (3) 17 (1967), 470–486. MR 217118, DOI 10.1112/plms/s3-17.3.470
- S. A. Amitsur, Identities in rings with involutions, Israel J. Math. 7 (1969), 63–68. MR 242889, DOI 10.1007/BF02771748
- Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
- Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099
- Wallace S. Martindale III, Rings with involution and polynomial identities, J. Algebra 11 (1969), 186–194. MR 234990, DOI 10.1016/0021-8693(69)90053-2
- Louis Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219–223. MR 309996, DOI 10.1090/S0002-9904-1973-13162-3
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- 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