Question

Let B = { f: ℝ  → ℝ | f is continuous } be the ring of all continuous functions from the real numbers to the real numbers. Let a be any real number and define the following function:

Φa:B→R

f(x)↦f(a)

It is called the evaluation homomorphism.

(a) Prove that the evaluation homomorphism is a ring homomorphism
(b) Describe the image of the evaluation homomorphism.
(c) Describe the kernel of the evaluation homomorphism.
(d) What does the First Isomorphism Theorem for Rings say in this example?
(e) Is the kernel, K, of the evaluation homomorphism a prime ideal or a maximal ideal or both or neither?

Answer 1