Volume 7, Issue 3 (Vol.7, No.3, 2021)                   mmr 2021, 7(3): 675-681 | Back to browse issues page


XML Persian Abstract Print


Download citation:
BibTeX | RIS | EndNote | Medlars | ProCite | Reference Manager | RefWorks
Send citation to:

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
, nazari.ar@lu.ac.ir
Abstract:   (737 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,R0. 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 =pAss Rq(p)  be a reduced primary decomposition of the zero ideal. Set  N=pmAss(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:RaRa.

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:RNm2.

3. Let R,m be a Noetherian local ring such that vRdimR+1. Then, we have ExtR1RRa,R0 for any principal ideal Ra of height one.

 
Full-Text [PDF 664 kb]   (139 Downloads)    
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

Add your comments about this article : Your username or Email:
CAPTCHA

Send email to the article author


Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

© 2024 CC BY-NC 4.0 | Mathematical Researches

Designed & Developed by : Yektaweb