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Extended centroids of power series rings

Published online by Cambridge University Press:  18 May 2009

W. S. Martindale ,
M. P. Rosen  and
J. D. Rosen
W. S. Martindale
Affiliation:
Department of MathematicsUniversity of Masschusetts at AmherstAmherst, Massachusetts 01003
M. P. Rosen
Affiliation:
Department of MathematicsCalifornia State University, NorthridgeNorthridge, California 91330
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Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 3 , September 1990 , pp. 371 - 375
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

1.Brewer, J. W., Power series over commutative rings, Lecture Notes in Pure and Appl. Math. 64 (Marcel Dekker, 1981).Google Scholar
2.Formanek, E., Maximal quotient rings of group rings, Pacific J. Math. 53 (1974), 109–116.CrossRefGoogle Scholar
3.Gilmer, R., Multiplicative ideal theory, Pure Appl. Math. 12 (Marcel Dekker, 1972).Google Scholar
4.Kharchenko, V. K., Galois theory of semiprime rings, Algebra i Logika 16 (1977), 313–363; English transl. (1978), 208–258.Google Scholar
5.Martindale, W. S., Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584.CrossRefGoogle Scholar
6.Martindale, W. S., The normal closure of the coproduct of rings over a division ring, Trans. Amer. Math. Soc. 293 (1986), 303–317.CrossRefGoogle Scholar
7.Matczuk, J., Extended centroids of skew polynomial rings, Math. J. Okayama Univ. 30 (1988), 13–20.Google Scholar
8.Passman, D. S., Infinite cross products, Pure Appl. Math. 135 (Academic Press, 1989).Google Scholar
9.Posner, E. C., Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), 180–184.CrossRefGoogle Scholar
10.Rosen, J. and Rosen, M., Extended centroids of skew polynomial rings, Canad. Math. Bull. 28 (1985), 67–76.CrossRefGoogle Scholar