标题： 幂闭合的理想在多项式和Laurent多项式环中 显示英文标题

标题： Power-closed ideals of polynomial and Laurent polynomial rings

摘要： 我们研究复数多项式环$R = \mathbb{C}[x_1,\ldots,x_d]$和广义多项式环$R^{\pm} = \mathbb{C}[x_1,\ldots,x_d]^{\pm} = M^{-1}\mathbb{C}[x_1,\ldots,x_d]$的幂闭合理想的结构，其中$M$是$R$的乘法子独异群$M = [x_1,\ldots,x_d]$。 这里，一个理想的$I$是{\em 幂闭合}，如果$f(x_1,\ldots,x_d)\in I$为每个自然数$i$都蕴含$f(x_1^i,\ldots,x_d^i)\in I$。特别地，我们研究了在$R$和$R^{\pm}$的理想集上的相关闭包和内部算子。 最后，我们给出了$R$和$R^{\pm}$的主幂闭合理想以及一般幂闭合理想的根的完整描述。 显示英文摘要

摘要： We investigate the structure of power-closed ideals of the complex polynomial ring $R = \mathbb{C}[x_1,\ldots,x_d]$ and the Laurent polynomial ring $R^{\pm} = \mathbb{C}[x_1,\ldots,x_d]^{\pm} = M^{-1}\mathbb{C}[x_1,\ldots,x_d]$, where $M$ is the multiplicative sub-monoid $M = [x_1,\ldots,x_d]$ of $R$. Here, an ideal $I$ is {\em power-closed} if $f(x_1,\ldots,x_d)\in I$ implies $f(x_1^i,\ldots,x_d^i)\in I$ for each natural $i$. In particular, we investigate related closure and interior operators on the set of ideals of $R$ and $R^{\pm}$. Finally, we give a complete description of principal power-closed ideals and of the radicals of general power-closed ideals of $R$ and $R^{\pm}$.