November 21, 2014

http://coursework.tylerlogic.com/courses/upenn/math508/homework10/bonusproblem
- B-1
- B-2
- Let be the ring of continuous functions on [0,1].
- a)
- Let c ∈ [0,1] be fix. Define a φ : → ℝ by
Proof. The map φ preserves the operations of addition and multiplication due to

and

Furthermore the additive and multiplicative identities of are mapped to the additive and multiplicative identities of ℝ:

and

__

With this definition of φ, we have that the kernel is exactly I = {f ∈ | f(c) = 0}. Hence with the above lemma, I is an ideal of . Futhermore, due to the above lemma and because φ is surjective (for a ∈ [0,1], the map xa is mapped to a by φ), the Fundamental Theorem of Ring Homomorphisms implies that ∕I is isomorphic to ℝ since ℝ is a field, this implies that the ideal I is a maximal ideal.

- b)