Question

Problem 8. Let R be the ring of all continuous functions from the closed interval [0, 1] to R andfor each c € [0, 1] let M₂ = {ƒ € R|f(c) = 0} (recall that Me was shown to be a maximalideal of R). (a) Prove that if M is any maximal ideal of R then there is a real number c € [0, 1] such that M = Mc. (b) Prove that if b and c are distinct points in [0, 1] then M₂ ‡ Me. ) Prove that Me is not equal to the principal ideal generated by - c. (d) Prove that Me is not a finitely generated ideal.

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