The specialization lemma in one of its forms (Prop. 6.2.7) states that in a field of infinite degree over its centre, itself infinite, there are no rational identities, and the proof depended on Amitsur's GPI-theorem. In 7.1 we again take up GPIs and examine their relation with ordinary polynomial identities.

The functional approach leads in 7.2 to another treatment of rational identities in fields, and the rational topology, a topic to which we shall return in Ch. 8. To study rational identities we need, besides the free field, the generic division algebras of different PI-degrees. They are introduced in 7.3; the specializations between them are described there and are illustrated in 7.4.

The rest of the chapter is devoted to an exposition of Bergman's theory of specializations. The basic notions of rational meet and support relation are explained in 7.5 and in 7.6 we see how they are realized in generic division algebras. Finally in 7.7 examples of the different support relations are given, showing the totally different behaviour in the non-commutative case.

Polynomial identities

Every ring satisfies certain identities such as the associative law: (xy)z = x(yz). In a field the situation is less simple; we have rational identities like xx-1 = 1 or (xy)-1 = y-1x-1, but here it is necessary to restrict x and y to be different from zero. In order to discuss rational identities over a field it is helpful first to summarize the situation for rings.