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Abstract algebra

  1. Jul 31, 2010 #1
    1. The problem statement, all variables and given/known data

    Let R be any ring and f:Z→R a homomorphism.

    a)Show that f is completely determined by the single value f(1)
    b)Determine all possible homomorphisms f in the case when R = Z.

    2. Relevant equations



    3. The attempt at a solution
    This question has me totally confused. I have gone through all the properties of homororphisms in the book but i am still confused.How is the homomorphism completely determined by one value anyway?
     
  2. jcsd
  3. Jul 31, 2010 #2
    The definition of homomorphism says that
    [tex]f(n)=n f(1)[/tex]
    and so knowledge of f(1) is enough to compute f for any integer.
     
  4. Jul 31, 2010 #3
    To summarize the above: f(1) determines f for every integer because the integers are generated by 1. Have you made any progress on part b?
     
  5. Jul 31, 2010 #4
    i cant find this property anywhere in my notes. Is this a property of homomorphismsm in general or only in the cas on the integers?
     
  6. Jul 31, 2010 #5
    Apply the defining property of a homomorphism with the domain being the group of integers. Does that make sense?
     
  7. Jul 31, 2010 #6

    Hurkyl

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    If figuring out the entire homomorphism from the value of f(1) is too hard, then try something simpler. What is f(0)? f(2)?
     
  8. Aug 1, 2010 #7
    but for a homomorphism we have:

    f(n)=f(1*n)=f(1)*f(n)
    thus f(1) =1
    so how is it completeley determined by f(1), which is always one anyway, i thought it would therefore be totally determined by the domain, depending on whether in is integers or rational numbers etc
     
  9. Aug 1, 2010 #8
    A ring has + as well as *, and the homomorphism must respect both. Use the fact that + is preserved as well to extract more information - such as f(2)
     
  10. Aug 1, 2010 #9
    ok thanks, using the additive property i see now that we get f(n)=nf(1) and it is determuned by this value. Is it sufficient for part b to use the multiplicative property, and the fact that the identity in z is 1 and thus f(1) maps to 1 to show that the only homomorphism is the identity map.Or is there something more in depth i could do?
     
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