Alireza Nazari,
Volume 7, Issue 3 (12-2021)
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.
