The Dilworth Number of Artinian Rings and Finite Posets with Rank Function
The Dilworth Number of Artinian Rings and Finite Posets with Rank Function
复制标题
Artinian环和具有秩函数的有限偏集的迪尔沃斯数
DOI：
10.2969/aspm/01110303
复制
发表时间：
1987
影响因子：
5.6
通讯作者：
J. Watanabe
中科院分区：
文献类型：
--
作者：
J. Watanabe
在文[13]中，我证明了对Artin局部环A的任意理想a和任意非单位元yin A，存在不等式μ（a）<l（A/yA）。因此，我们自然地考虑数d（A）：= Max {μ（A）}和r（A）：= Min {l（A/yA）}。本文有两个目的：（1）给出数d（A）的组合解释;（2）研究等式d（A）= r（A）成立的一种情况。虽然关于理想生成元个数的问题已经引起了相当大的关注（例如Sally [9]），但Artin环A的个数d（A）似乎从未被明确考虑过。但是，只要一个试图计算的数字，采取的例子阿廷环的"单项式型"，人们意识到，这是一个相当组合的问题，幸运的是，一些定理和某些想法在组合可用于目的。其中包括Dilworth定理、Sperner性质和偏序集的对称链分解。我称d（A）为Artin环A的Dilworth数，因为对于单名型的Artin环A，偏序集自然是相关联的，并且d（A）与组合学家所称的偏序集的Dilworth数一致。关于数r（A），我称它为Rees数，因为Rees [8]定义了局部环的一般元素的概念。文[13]和本文所采用的Artinian情形下的一般元的定义与他的定义略有不同：即我们说y是A的一般元，如果l（A/yA）= r（A），且A有无限剩余域。这个数的意义在于它限制了环的理想的生成元的个数。也就是说，d（A）<r（A）。现在，一个自然的问题出现了：什么时候平等成立？要回答这个问题似乎很困难，因为此时还不能期望一个普遍的理论，我们在这里所做的就是考虑某一类
In my paper [13] I proved that for any ideal a of an Artinian local ring A and for any non-unit element yin A there is an inequality μ(a)< l(A/yA). Thus we are naturally led to consider the numbers d(A):=Max{μ(a)} and r(A):=Min {l(A/yA)}. The present paper has two purposes: (1) To give a combinatorial interpretation of the number d(A) and (2) to study one case where the equality d(A)=r(A) holds. Although the problems concerning the number of generators of ideals have drawn considerable attention (for examples Sally [9]), the number d(A) of an Artinian ring A does not seem to have ever been considered explicitly. But as soon as one tries to compute the number, taking an example of Artinian ring of "monomial type", one realizes that this is quite a combinatorial question, and fortunately some theorems and certain ideas in combinatorics are available for the purpose. To mention some of these, Dilworth's theorem, Sperner property and symmetric chain decomposition of posets. I called the number d(A) the Dilworth number of the Artinian ring A because with an Artinian ring A of monomial type a poset is naturally associated and d(A) coincides with what the combinatorists call the Dilworth number of the poset. As to the number r(A), I called it the Rees number because Rees [8] defined the notion of general elements of local rings in a general setting. The definition of a general element in the Artinian case adopted in [13] and in the present paper is slightly different from his: namely we say that y is a general element of A if l(A/yA)=r(A) provided that A has an infinite residue field. The significance of this number is that it bounds the number of generators of ideals of the ring. I.e., d(A)<r(A). Now a natural question arises: when does the equality hold? To answer this question seems very difficult, and because a general theory cannot be expected at this time, what we do here is to consider a certain class of