Pairs of Rings Whose All Intermediate Rings Are G–Rings

A G–ring is any commutative ring R with a nonzero identity such that the total quotient ring $$\mathbf {T}(R)$$ is finitely generated as a ring over R. A G–ring pair is an extension of commutative rings $$A\hookrightarrow B$$ , such that any intermediate ring $$A\subseteq R\subseteq B$$ is a G–ring. In this paper we investigate the transfer of the G–ring property among pairs of rings sharing an ideal. Our main result is a generalization of a theorem of David Dobbs about G–pairs to rings with zero divisors.