ON INTERMEDIATE RINGS WHICH ARE FINITELY GENERATED MODULES OVER A NOETHERIAN RING

Abstract

Let (R, m) be a commutative Noetherian ring and Q(R) the total quotient ring of R. The aim of this paper is to study the
structure of intermediate rings between R and Q(R). Let X be
the set of all equivalent classes [I], where I is an ideal of R such
that I 2 = aI for some non zero divisor a ∈ I. Let Y be the set
of all intermediate rings A between R and Q(R) such that A
is finitely generated R-modules. In this paper, we establish a
bijection from X to Y. Some examples are given to clarify the
result. Firstly, we show that if R is a principal ideal domain,
then R is the unique element of Y. Secondly, we give a Buchsbaum ring R which is not Cohen-Macaulay and we construct a
Cohen-Macaulay intermediate ring A ∈ Y. In order to solve the
problem, we apply the method investigated by S. Goto in 1983,
L. T. Nhan and M. Brodmann 2012.