# A question about ring homomorphisms

1. Jan 1, 2013

### Artusartos

1. The problem statement, all variables and given/known data

If R is a domain with F=Frac(R), prove that Frac(R[x]) is isomorphic to F(x).

2. Relevant equations

3. The attempt at a solution

Let $$\phi : Frac(R[x]) \rightarrow F(x)$$ be a map sending (f(x),g(x)) to f(x)/g(x). We need to show that $$\phi$$ is a ring homomorphism. Let f,g,h,k be in R[x] such that f/h and g/k is in Frac(R[x]).

We know that

$$\phi(1,1) = 1/1= 1$$

$$\phi (fg, hk) = \frac{fg}{hk} = \frac{f}{h}\frac{g}{k} = \phi(f,h)\phi(g,k)$$

But I'm confused with the addition part...

$$\phi(f+g,h+k) = \frac{f+g}{h+k}$$

$$\phi(f,h)+\phi(g,k) = \frac{f}{h}+\frac{g}{k} = \frac{kf+gh}{h+k}$$

But now $$\phi(f+g,h+k) \not= \phi(f,h) + \phi(g,k)$$

Can anybody help with this?

2. Jan 1, 2013

### HallsofIvy

You appear to be interpreting "Frac[R(x)]" as pairs, (f(x), g(x)). How are you defining addition in Frac[R(x)]?

3. Jan 1, 2013

### Artusartos

Oh...ok I think I get what you mean...because $$\phi(f,h + g,k) = \phi(fk+gh, hk)$$, right?

Last edited: Jan 1, 2013