Ideal and maximal ideal
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Ideal and maximal ideal
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.
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.
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