A Theorem of Martindale Revisited by the Local Approach
by
Maria Isabel Tocon Barroso
Universidad de Málaga

Algebras satisfying a polynomial identity seem to be the natural generalization of commutative algebras: A is said to satisfy a polynomial identity over K, or just to be a PI algebra, if there exists a nonzero f in K < x₁, ..., x_n > , the free algebra over K in the noncommutative variables x₁, ..., x_n, for some n, such that f(a₁, ..., a_n)=0 for all a₁, ..., a_n in A.

Classical results on PI algebras are the theorems of Kaplansky and Posner describing primitive and prime PI algebras, respectively. Both theorems can be obtained as a consequence of a theorem of Martindale which characterizes the prime algebras satisfying a "generalized polynomial identity" (GPI) [1].

The aim of this talk is to obtain the characterization given by Martindale's theorem, looking not at the PI character of the algebra, but at its "local PI structure" (following the ideas of [3]). This local-to-global transition of information will be possible via the "local algebras" associated to elements, which were first introduced by K. Meyberg [2]:

Let A be an algebra. For every element a in A, take A^(a) to be the algebra defined by the same linear structure as A and the homotope product x._ay=xay. We define the local algebra of A at a as the quotient:

Aa = A(a)/ ker(a)

where ker(a)={x in A: axa=0}.

In particular, we will show that the condition on a prime algebra of satisfying a GPI is equivalent to that of containing a nonzero PI element, where an element a in A is said to be PI if the local algebra A_a is PI (indeed, both notions are equivalent in practice). Next, we state the main result, which is a local version of Martindale's theorem:

Theorem Let A be a prime K-algebra and a in A a nonzero PI element. Then:

(i) The extended centroid C(A) of A is isomorphic to the field of fractions of the center Z(A_a) of A_a.

(ii) The central closure of A is a primitive algebra with nonzero socle, whose associated division algebra D has finite dimension over its center C(A).

References

[1] W.S. Martindale, Prime rings satisfying a Generalized Polynomial Identity J. Algebra 12, 1969, 576-584

[2] K. Meyberg, Lectures on algebras and triple systems Lecture Notes, University of Virginia, Charlottesville 1972

[3] F. Montaner, Local PI Theory of Jordan Systems J. Algebra 216, 1999, 302-327

Date received: March 12, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cage-16.