Ideals In \(\mathcal{C}[0,1]\)

Discrete Mathematics Level 4

An ideal in \(\mathcal{C}[0,1]\) is a subring \(I \subset \mathcal{C}[0,1]\) such that for any \(f\in I\) and \(g \in \mathcal{C}[0,1]\), the product \(fg\) is in \(I\).

If \(I \subset \mathcal{C}[0,1]\) is an ideal such that \(I \neq \mathcal{C}[0,1]\), then \(I\) is called proper.

A maximal ideal \( M \subset \mathcal{C}[0,1]\) is a proper ideal such that whenever \(I\) is an ideal with \(M \subset I\), then in fact \(M = I\).

(In other words, \(M\) is maximal with respect to the partial order given on ideals of \(\mathcal{C}[0,1]\) by set inclusion).

For \(x\in [0,1]\), define \[M_x := \{f \in \mathcal{C}[0,1] \, | \, f(x) = 0\}.\] Note that \(M_x\) is an ideal in \(\mathcal{C}[0,1]\).

Answer the following yes-no questions:

  • Is \(M_x\) is a maximal ideal of \(\mathcal{C}[0,1]\) for all \(x\in [0,1]\)?
  • Does there exist a maximal ideal of \(\mathcal{C}[0,1]\) that is not of the form \(M_x\) for some \(x\in [0,1]\)?

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