[MATH113-Abstract Algebra]证明在[0,1]上连续函数集的在某点不为0的理想为这个函数集本身

题目

1. Let R denote the ring of continuous functions from the unit interval [0,1] to R. Let I be an ideal of
   R with the property that for every point x of [0,1], there is a function f ∈ I with f(x) ≠ 0. Show I = R.
   (Hint: The unit interval is compact.）

证明

我们由区间内的某个点找到一个对应的函数然后根据这个函数的连续性,得到一个在x附近的领域，最后根据compact的特性得出结论