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Nazari A. Some results on the vanishing of certain Ext-modules. mmr 2021; 7 (3) :675-681
URL: http://mmr.khu.ac.ir/article-1-2982-en.html
Some results on the vanishing of certain Ext-modules
Alireza Nazari *
, nazari.ar@lu.ac.ir
Abstract:   (1117 Views)
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
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Type of Study: Original Manuscript | Subject: alg
Received: 2019/07/22 | Revised: 2023/06/18 | Accepted: 2019/12/11 | Published: 2021/12/1 | ePublished: 2021/12/1
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