Some results on the vanishing of certain Ext-modules

Author
Alireza Nazari
10.52547/mmr.7.3.675
Abstract
Introduction

Let R be a commutative Noetherian ring with identity, and let a be an element of R. Assume that either (a) R is an integral domain, or (b) R is a Cohen–Macaulay ring. It is well known that if ht Ra=1, then ExtR1RRa,R≠0. So, it is natural to ask the following questions:

Question 1: Let R be a commutative Noetherian ring with identity, and a be an element of R such that ht Ra=1. Then, is it true that, ExtR1RRa,R≠0?

or in more general situations,

Question 2: Let R be a commutative Noetherian ring with identity, and a be an element of R. Under what conditions is ExtR1RRa,R≠0?

The aim of this paper is to answer these questions.

Results and discussion

In this paper, first we show that ExtR1RRa,R≅∘:R∘:RaRa and next we give some conditions which guarantee that ExtR1RRa,R≠0. In addition we give some examples to illustrate these results. We denote the set of minimal members of associated primes of R by mAss(R). Let 0 =p∈Ass Rq(p) be a reduced primary decomposition of the zero ideal. Set N=p∈mAss(R)q(p).

The following conclusions were drawn from this research.

1. Let R be a Noetherian ring and a be an element of R such that ht Ra=1. Then, ExtR1RRa,R≠0, if either of the following conditions holds:

a) aN=0;

b) a2N=0 and 0:Ra⊆Ra.



2. Let R,m be a Noetherian local ring and a be an element of R such that ht Ra=1. Then we have ExtR1RRa,R≠0, if either of the following holds:

a) N2=0;

b) 0:RN⊈m2.



3. Let R,m be a Noetherian local ring such that vR≤dimR+1. Then, we have ExtR1RRa,R≠0 for any principal ideal Ra of height one.


Keywords
Noetherian rings
Ext-modules
principal ideals
1. Bruns W., Herzog J., “Cohen-Macaulay Rings”, Cambridge University Press, Cambridge (1997).

2. Matsumura H., “Commutative Ring Theory”, Cambridge University Press, Cambridge (1989).

3. Rotman J. J., “An Introduction to Homological Algebra”, Academic Press, New York (1979).
Mathematical Research
Volume 7, Issue 3 - Serial Number 18
Autumn 2021
Pages 675-681