Ideal and maximal ideal

[Papapetros Vaggelis]

Let \(\displaystyle{\left(C(\left[0,1\right]),+,\cdot\right)}\) be the commutative ring with unity, of all continuous



functions \(\displaystyle{f:\left[0,1\right]\longrightarrow \mathbb{R}}\) .

For each \(\displaystyle{X\in\mathbb{P}(\left[0,1\right])-\left\{\varnothing\right\}}\), define the set

\(\displaystyle{I_{X}=\left\{f\in C(\left[0,1\right]): f(x)=0\,,\forall\,x\in X\right\}}\) .



1. Prove that the set \(\displaystyle{I_{X}}\) is an ideal of the ring \(\displaystyle{\left(C(\left[0,1\right]),+,\cdot\right)}\)

for each \(\displaystyle{X\in\mathbb{P}(\left[0,1\right])-\left\{\varnothing\right\}}\).

2. If \(\displaystyle{X=\left\{x_0\right\}\subseteq \left[0,1\right]}\) for some \(\displaystyle{x_0\in\left[0,1\right]}\), then prove that

the ideal \(\displaystyle{I_{X}}\) is maximal.