A note on G-type rings

In this article, we introduce and examine the concept of G-type rings. A ring R is classified as a G-type ring if its total quotient ring, denoted by Q, is generated by a countable number of elements over R as an R-algebra. Formally, Q = R[S^{-۱}], where S is a countable set of regular elements in R. We establish that R is G-type if and only if there exists a countable set of regular elements, denoted as S, such that every prime ideal disjoint from S consists solely 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.

کلیدواژه ها:

Total quotient ring ، G-type ring ، G-type ideal ، Caliber ، Strongly algebraic