GENERALIZED RATIONAL IDENTITIES

Publisher Summary This chapter discusses the generalized rational identities. It presents a simple proof of the Amitsur–Bergman result with a somewhat different condition on the centers (which are still assumed infinite). The main tool is the notion of a universal skew field of fractions. This was constructed by Amitsur in the special case of free algebras, using precisely his results on rational identities. The explicit construction for universal skew fields of fractions to prove the result on rational identities is used. If R is any ring, then by afield of fractions of R, one understands a field K with an embedding R → K such that K is the field generated by the image. In the commutative case, such a K exists if R is an integral domain, and it is then unique. In general no necessary and sufficient conditions are known for a field of fractions to exist and even when it does exist, it need not be unique.