Is Y a Surjective Ring Homomorphism?

[cagoodm]

Homework Statement

Let T be the ring of all continuous functions from the closed interval [0,1] to R. Define a function Y: T-->R by Y(f) = f(1)

Prove that Y is a surjective ring homomorphism.

Is Y an isomorphism? Prove your answer is correct.I honestly just don't understand what this is asking... but I think it has to be surjective because every function in f is mapping to f(1). Is that the right way of thinking about it?

But, it can't be an isomorphism because it's not one to one, there are multiple functions mapping to the same f(1)

Any help would be greatly appreciated. Thank you!

 

[micromass]

Well, you're doing the correct things. The only thing you still got to do is to write it formally.

Let's begin with the homomorphism thingy. You need to show that
1) Y(f+g)=Y(f)+Y(g)
2) Y(fg)=Y(f)Y(g)
3) Y(1)=1


Then for surjectivity. You need to take an a in R arbitrarly. You'll need to find a function f such that Y(f)=a. There's an easy function f which does that.