Let R=C([0,1]) be the ring of continuous real-valued functions on the interval [0,1]. Let T⊆[0,1], and let
I(T)={f∈R:f(x)=0∀x∈T} .
(i) Prove that I(T) is an ideal of R.
(ii) If x∈[0,1] and mx=I({x}), show that R/Mx≅R, and hence Mx is a maximal ideal of R.
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