Comm. Alg. 29 (4), 1659 - 1668 (2001)
 

Abstract.
In this note we give a sufficient condition for an artinian ring R  to be an artin algebra:  R  satisfies a polynomial identity and the diagram  D  of  R is of type A, i. e. each connected component  D'  of D  is a tree and satisfies the condition that the product of all valuations of the edges in  D'  is one or a prime number.   Examples of diagrams of type  A   are the Dynkin diagrams and the Euclidean diagrams of type  BCn~, CDn~, Dn~, E6~, E7~, E8~, F41~, F42~, G21~ and G22~.  Moreover, for each diagram  D  which is not of type A  we construct an artinian PI-ring which has diagram  D and which is not an artin algebra.

Our main tools are an application of a Lemma of P. Dowbor on skew field extensions, and the construction of pairs of automorphisms of fields which have finite order such that their composition has a small fixed field.

Download  aaa.dvi  or  aaa.ps .

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Last modified:  September 26, 2001, by Markus Schmidmeier