نتایجی در مورد صفر شدن برخی ext مدولهای خاص

فرض کنید r یک حلقه ی جابه جایی یک دار نوتری و a عضوی از r باشد.در این مقاله، به بررسی و مطالعه ی شرایطی روی حلقه ی r و ایده آل اصلی ra می پردازیم که تحت آن داشته باشیم: 0≠(r/ra,r)ext1r .

some results on the vanishing of certain ext modules

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.