Email this article ON INTERMEDIATE RINGS WHICH ARE FINITELY GENERATED MODULES OVER A NOETHERIAN RING
About this article
Received: 07/01/21 Revised: 10/03/21 Published: 22/03/21Authors
1. Nguyen Xuan Linh, National University of Civil Engineering2. Luu Phuong Thao
, TNU - University of Education
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.
Keywords
Full Text:
PDFReferences
[1] S. Goto, "On the CohenMacaulayfication of certain Buchsbaum rings," Nagoya Mathematical Journal, vol. 80, pp. 107-116, 1980.
[2] P. Schenzel, "On birational Macaulayfications and Cohen–Macaulay canonical modules," Journal of Algebra, vol. 275, pp. 751-770, 2004.
[3] T. N. An, L. T. Nhan, and L. P. Thao, "Non Cohen-Macaulay locus of canonical modules," Journal of Algebra, vol. 525, pp. 435-453, 2019.
[4] M. Brodmann and L. T. Nhan, "On canonical Cohen-Macaulay modules," Journal of Algebra, vol. 371, pp. 480-491, 2012.
[5] N. V. Trung, “Toward a theory of generalized Cohen- Macaulay modules,” Nagoya Mathematical Journal, vol. 102, pp. 1-49, 1986.
DOI: https://doi.org/10.34238/10.34238/tnu-jst.3888
Refbacks
- There are currently no refbacks.