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Azumaya algebras and rings with polynomial identity

Published online by Cambridge University Press:  24 October 2008

Alfred Goldie
Alfred Goldie
Affiliation:
University of Leeds
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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

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References

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