Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-04T18:05:52.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Azumaya algebras and rings with polynomial identity

Published online by Cambridge University Press:  24 October 2008

Alfred Goldie
Alfred Goldie
Affiliation:
University of Leeds
Get access Rights & Permissions [Opens in a new window]

Extract

The Formanek polynomial gives an explicit expression for the calculation of central elements in a prime ring with polynomial identity, in such a form as to provide a set of linear maps of the ring into its centre. This note makes use of properties of these maps with the object of giving a direct proof of the remarkable theorem of M. Artin on the relationship between Azumaya algebras and rings with polynomial identities. We show that such a central map can be chosen to have some of the crucial properties of the reduced trace in matrices, notably that of non-degeneracy, and our proof of the theorem rests on these properties. The method can also be used for the study of prime PI rings in general; this is illustrated by giving new proofs of theorems of Formanek and Cauchon. The changes of proof essentially amount to the elimination of the need for a change of rings to obtain a split extension in simple algebras. The main theorem is the following.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 3 , May 1976 , pp. 393 - 399
Copyright
Copyright © Cambridge Philosophical Society 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Amitsur, S. A.Polynomial identities and Azumaya algebras. J. Algebra 27 (1973), 117–125.CrossRefGoogle Scholar
(2)Artin, M.On Azumaya algebras and finite dimensional representations of rings. J. Algebra 11 (1969), 532–563.CrossRefGoogle Scholar
(3)Bass, H.Lectures on topics in algebraic K-theory. Tata Institute, Bombay, 1967.Google Scholar
(4)Bourbaki, N.Algèbre commutative, chapters 1–2 (XXVII) (Herman, Paris, 1961).Google Scholar
(5)De Meyer, F. and Ingraham, E.Separable algebras over commutative rings, Lecture notes No. 181 (Springer, 1971).CrossRefGoogle Scholar
(6)Formanek, E.Central polynomials for matrix rings. J. Algebra 23 (1972), 129–132.CrossRefGoogle Scholar
(7)Kaplansky, I.Commutative rings (Allyn and Bacon, Boston, 1970).Google Scholar
(8)Procesi, C.On a theorem of M. Artie. J. Algebra 22 (1972), 309–315.CrossRefGoogle Scholar
(9)Row, L. Thesis (Yale University).Google Scholar