G-DOMAINS, G~IDEALS, AND HILBERT RINGS APPROVED:

The problem with which this investigation is concerned is that of determining the properties of the following; a particular type of integral domain, the G-domain ' a type of prime ideal, the G-idealj and a special type of ring, the Hilbert ring. An integral domain R with a quotient field K is said to be a G-domain if K, as a ring, is finitely generated over R. From the definition of a G-domain, it follows that if the intersection of all non-zero prime ideals of an integral domain with a quotient field is non-zero, then the integral domain is a G-domain. Conversely, in a G-domain the inter-section of all non-zero prime ideals is non-zero. Also, if R is a G-domain,then R[x], where x is an indeterminate, is never a G-domain. Let R be a ring with unity and u be an element of a larger domain, then if R[x] is a G-domainJ th