The ratliff-rush ideals in a noetherian ring
The ratliff-rush ideals in a noetherian ring
复制标题
诺特环中的拉特利夫冲理想
DOI：
10.1080/00927879208824359
复制
发表时间：
1992
影响因子：
0.7
通讯作者：
Kishor Shah
中科院分区：
文献类型：
--
作者：
W. Heinzer;David Lantz;Kishor Shah
Ratliff和Rush特别证明了，对于足够大的正整数n，（）= I，是最大理想，因此I =。我们称正则理想I为I = Ratliff-Rush理想，我们称与I相关的Ratliff-Rush理想为Ratliff-Rush理想。很容易看出，I：I中的元素a在I上是积分的，在这个意义上，存在一个形式为a + b1 a k−1 +的方程。. . + bk = 0，其中bi ∈ I，i = 1，. . .，k.因此，理想的理想总是在I和I的积分闭包I ′之间，因此积分闭理想是Ratliff-Rush理想。Ratliff和Rush观察到[RR，（2.3.4）]可逆理想的幂是Ratliff-Rush理想，因此任何由非零因子生成的主理想都是Ratliff-Rush理想。他们还证明了一个有趣的事实，即对于R的任何正则理想I，存在正整数n，使得对所有k ≥ n，| k = I [RR，（2.3.2）]，即，正则理想的所有充分高的幂都是Ratliff-Rush。一个正则理想I总是其相关的Ratliff-Rush理想的约化，在这个意义上，对于某个正整数n，I（）=（）。关于理想的约化和约化数的基本事实，我们请读者参阅[NR]、[H1]和[H2]。特别地，如果存在理想I的一个元素a，对于该理想I，aR = I，则aR称为I的主约化，并且该方程成立的最小n称为I的约化数。我们称一个正则理想I稳定当且仅当它有一个约化数至多为1的主约化，即，当且仅当存在I的元素a，
Ratliff and Rush show in particular that Ĩ is the largest ideal for which, for sufficiently large positive integers n, (Ĩ) = I and hence that ̃̃ I = Ĩ. We call regular ideals I for which I = Ĩ Ratliff–Rush ideals, and we call Ĩ the Ratliff–Rush ideal associated with I. It is easy to see that an element a of I : I is integral over I, in the sense that there is an equation of the form a + b1a k−1 + . . . + bk = 0, where bi ∈ I for i = 1, . . . , k. Therefore, the ideal Ĩ is always between I and the integral closure I ′ of I, and hence integrally closed ideals are Ratliff–Rush ideals. Ratliff and Rush observe [RR, (2.3.4)] that the powers of an invertible ideal are Ratliff–Rush ideals, so any principal ideal generated by a nonzerodivisor is a Ratliff–Rush ideal. They also prove the interesting fact that for any regular ideal I of R, there is a positive integer n such that for all k ≥ n, Ĩk = I [RR, (2.3.2)], i.e., all sufficiently high powers of a regular ideal are Ratliff–Rush. A regular ideal I is always a reduction of its associated Ratliff–Rush ideal Ĩ, in the sense that I(Ĩ) = (Ĩ) for some positive integer n. For the basic facts on reductions and reduction numbers of ideals, we refer the reader to [NR], [H1], and [H2]. In particular, if there is an element a of an ideal I for which aI = I then aR is called a principal reduction of I and the smallest n for which this equation holds is called the reduction number of I. We will call a regular ideal I stable iff it has a principal reduction with reduction number at most one, i.e., iff there is an element a of I for which