PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I)
PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I)
复制标题
单项理想的参数分解（一）
DOI：
--
复制
发表时间：
--
期刊：
影响因子：
--
通讯作者：
Kishor Shah
中科院分区：
文献类型：
--
作者：
W. Heinzer;L. Ratliff;Kishor Shah
Emmy Noether证明了Notherian环中的每个理想都允许分解为不可约理想。在本文中，我们在一种基本情况下显式地计算这种分解。具体地说，设R是有单位元的交换环，x1，.。。，Xd(d>；1)是R-序列，设X=(x1，.。。，Xd)R，设I是单项理想(即，由单项式x11生成的真理想)，使得Rad(I)=Rad(X)。然后，主要结果给出了作为形如(x11，.)的理想的无冗余有限交的I的规范唯一分解。。。，x和d)R，其中指数n1，.。。，d为正整数。具体地说，如果Z1，.。。，zm是(i：x)−i中的单项式，如果zj=xaj，1−11··xaj，d−1d，则i=∩{(xj，11，.。。，xAj，d d)R；j=1，.。。、m}。我们还计算了由i的单项生成元的k次方生成的理想i[k]的分解。我们使用的方法是代数的，但它们是由格子的几何学提出的。
Emmy Noether showed that every ideal in a Noetherian ring admits a decomposition into irreducible ideals. In this paper we explicitly calculate this decomposition in a fundamental case. Specifically, let R be a commutative ring with identity, let x1, . . . , xd (d > 1) be an R -sequence, let X = (x1, . . . , xd)R, and let I be a monomial ideal (that is, a proper ideal generated by monomials x1 1 · · · x ed d ) such that Rad(I) = Rad(X). Then the main result gives a canonical and unique decomposition of I as an irredundant finite intersection of ideals of the form (x1 1 , . . . , x nd d )R, where the exponents n1, . . . , nd are positive integers. Specifically, if z1, . . . , zm are the monomials in (I : X)− I , and if zj = x aj,1−1 1 · · ·x aj,d−1 d , then I = ∩{(xj,1 1 , . . . , x aj,d d )R; j = 1, . . . ,m}. We also calculate the decomposition of the ideals I [k] generated by the k -th powers of the monomial generators of I . The methods we use are algebraic, but they were suggested by the geometry of lattices.