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

MathSciNet review:
0323822

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Abstract: Let denote a ring with involution , where ``involution'' means `` ". We can specialize many ring-theoretical concepts to rings with involution; in particular an *ideal* of is an ideal of stable under , and *the center of* is the set of central elements of which are fixed under . Then we say *is prime* when the product of any two nonzero ideals of is nonzero; similarly *is semiprime* when any power of a nonzero ideal of is nonzero. The main result of this paper is a strong analogue to Posner's theorem [**5**], namely that any prime with polynomial identity has a ring of quotients , formed merely by adjoining inverses of nonzero elements of the center of . This quotient ring is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.

**[1]**S. A. Amitsur,*Prime rings having polynomial identities with arbitrary coefficients*, Proc. London Math. Soc. (3)**17**(1967), 470–486. MR**0217118****[2]**S. A. Amitsur,*Identities in rings with involutions*, Israel J. Math.**7**(1969), 63–68. MR**0242889****[3]**Nathan Jacobson,*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR**0222106****[4]**Nathan Jacobson,*Structure and representations of Jordan algebras*, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR**0251099****[5]**Wallace S. Martindale III,*Rings with involution and polynomial identities*, J. Algebra**11**(1969), 186–194. MR**0234990****[6]**Louis Rowen,*Some results on the center of a ring with polynomial identity*, Bull. Amer. Math. Soc.**79**(1973), 219–223. MR**0309996**, 10.1090/S0002-9904-1973-13162-3

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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