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A question about ring homomorphisms

  1. Jan 1, 2013 #1
    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 [tex]\phi : Frac(R[x]) \rightarrow F(x)[/tex] be a map sending (f(x),g(x)) to f(x)/g(x). We need to show that [tex]\phi[/tex] 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

    [tex]\phi(1,1) = 1/1= 1[/tex]

    [tex]\phi (fg, hk) = \frac{fg}{hk} = \frac{f}{h}\frac{g}{k} = \phi(f,h)\phi(g,k)[/tex]

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

    [tex]\phi(f+g,h+k) = \frac{f+g}{h+k}[/tex]

    [tex]\phi(f,h)+\phi(g,k) = \frac{f}{h}+\frac{g}{k} = \frac{kf+gh}{h+k}[/tex]

    But now [tex]\phi(f+g,h+k) \not= \phi(f,h) + \phi(g,k)[/tex]

    Can anybody help with this?

    Thanks in advance
     
  2. jcsd
  3. Jan 1, 2013 #2

    HallsofIvy

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    Staff Emeritus
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    You appear to be interpreting "Frac[R(x)]" as pairs, (f(x), g(x)). How are you defining addition in Frac[R(x)]?
     
  4. Jan 1, 2013 #3
    Oh...ok I think I get what you mean...because [tex]\phi(f,h + g,k) = \phi(fk+gh, hk)[/tex], right?
     
    Last edited: Jan 1, 2013
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