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

Probability Level 4

An ideal in C[0,1]\mathcal{C}[0,1] is a subring IC[0,1]I \subset \mathcal{C}[0,1] such that for any fIf\in I and gC[0,1]g \in \mathcal{C}[0,1], the product fgfg is in II.

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

A maximal ideal MC[0,1] M \subset \mathcal{C}[0,1] is a proper ideal such that whenever II is an ideal with MIM \subset I, in fact M=IM = I. ((In other words, MM is maximal with respect to the partial order given on ideals of C[0,1]\mathcal{C}[0,1] by set inclusion.))

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

Answer the following yes-no questions:

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