Pairs of Rings Whose All Intermediate Rings Are G–Rings Lahoucine Izelgue and Omar Ouzzaouit Abstract A G–ring is any commutative ring R with a nonzero identity such that the total quotient ring T( R) is finitely generated as a ring over R. A G–ring pair is an extension of commutative rings A → B , such that any intermediate ring A ⊆ R ⊆ 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. Keywords G–domain · G–ring · G–ring pair · Amalgamated duplication 1 Introduction All rings considered in this paper are commutative with unit. An integral domain R is said to be a G–domain if the quotient field K of R is a finitely generated ring over R. This is equivalent to saying that the quotient field K is of the form R  1 t  for some nonzero element t ∈ R (cf. [6, Theorem 18]). An integral domain with only finitely many prime ideals is a G–domain. However, the polynomial ring with coefficients in R is never a G–domain [6]. Notice also that an infinite G–domain R has the same cardinality as its set of units U ( R) [2]. Thus, any infinite ring with a finite group of units, such as Z, is not a G–domain. On the other hand, Adams [1], introduced the concept of a G–ring as a generaliza- tion of Kaplansky’s definition of a G–domain to rings with zero divisors. He defined a G–ring to be any commutative ring R, with a nonzero identity, such that the total quotient ring T( R) is finitely generated as a ring over R. He then pointed out that T( R) is finite over R if, and only if, T( R) = R[u −1 ], for some regular element u in L. Izelgue · O. Ouzzaouit (B) Faculty of Sciences Semlalia, Department of Mathematics, Cadi Ayyad University, P.O. Box 2390, 40.000 Marrakech, Morocco e-mail: ouzzaouitomar@gmail.com L. Izelgue e-mail: izelgue@uca.ac.ma © Springer International Publishing AG 2018 A. Badawi et al. (eds.), Homological and Combinatorial Methods in Algebra, Springer Proceedings in Mathematics & Statistics 228, https://doi.org/10.1007/978-3-319-74195-6_11 111