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Show that a f: Z -> R , n -> n*1(subr) is a homomorphism of rings

  1. Nov 7, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that a f: Z → R , n → n*1R is a homomorphism of rings

    2. Relevant equations

    3. The attempt at a solution
    I'm not sure how to exactly go about answering this question, but I'm going to try to start with the definition:

    f(a+b) = f(a) + f(b)
    f(a*b) = f(a) * f(b)
    f(1Z) = 1R

    Ok, so if I actually have a clue what the problem statement means -- which is just slightly possible -- then this function maps values 'n' to 'n' times the identity unit of R. Which is equal to the identity unit of Z, so I think that part of the definition is more-or-less covered.

    How do I go about using the first two parts of the definition?

    The function is just f(n) = n right?
    so do I just sub in (a+b) for n?
    f(a+b) = f(a) + f(b) ???
    Is there anyway I can actually show that this is the case? I mean isn't this intuitive/properly defined upon the integers?

    same for the other part
    f(ab) = f(a)*f(b) ..... i'm just not sure how to actually show this...

    thanks for any help.
  2. jcsd
  3. Nov 7, 2011 #2


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    Science Advisor

    no, f(n) = n is't true.

    for example, if R = Z6, f(23) = 5.

    realize that n*1R isn't "n", it's:

    1R+1R+.....+1R (n summands).
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