Some Useful Notes on G-type Rings

In this article, we introduce and study the notion of G-type rings (note, a ring R is a G-type ring if its total quotient ring,Q say, is generated by a countable number of elements over R, as an R-algebra,that is to say Q=R[S^{-1}], where S is a countable set of regular elements in R. We observe that R is G-type if and only if there exists a countable set of regular elements in R, S say, such that every prime ideal which is disjoint from S consists of zero-divisors. It is shown that whenever a ring T is countably generated over a subring R, as an R-algebra, and T is strongly algebraic over R, then R is G-type if and only if T is G-type.