ON G-TYPE DOMAINS

Abstract

In this paper, we introduce and study the notion of G-type domains (a domain R is G-type if its quotient field is countably generated R-algebra). We extend some of the basic properties of G-domains to G-type domains. It's observed that a prime ideal of R[x1, x2,…,xn,…] is G-type if and only if its contractions in R, R[x1, x2,…,xn] for all n ≥ 1 are G-type. Using this concept we give a natural proof of the well-known Hilbert Nullstellensatz in infinite countable-dimensional spaces. Characterizations of Noetherian G-type domains, Noetherian G-type domains with the countable prime avoidance property are given. As a consequence, we observe that in complete Noetherian semi-local rings, G-type ideals and G-ideals are the same. Rings with countable Noetherian dimension which are direct sum of G-type domains are fully determined. Finally, we characterize Noetherian rings in which G-type ideals are maximal.