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There are no ring homomorphisms from Z5 to Z7

  1. Oct 29, 2006 #1
    I just need confirmation.

    I have a problem in my algebra class that says:
    Prove that there are no ring homomorphisms from Z5 to Z7.
    I have the following definition of ring homomorphism:
    Let R and S be rings. A function R to S is a ring homomorphism if the following holds:
    f(1R)=1S.
    f(r1+r2)=f(r1)+f(r2) for all r1 and r2 in R.
    f(r1r2)=f(r1)f(r2) for all r1 and r2 in R.

    I've been thinking and wouldn't f(x)=0 work?
    This is a problem in a published textbook so it doesn't make sense to me. Could anyone clue me in to where there might be a contradiction in the definition?

    Thanks in anticipation. Mike.
     
  2. jcsd
  3. Oct 29, 2006 #2

    StatusX

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    Then the first condition, that 1 maps to 1, isn't satisfied.
     
  4. Oct 29, 2006 #3
    Could you just explain why it isn't satisfied? I think I'm missing something.
     
  5. Oct 29, 2006 #4

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    You're saying f(1)=0, and 0 is not 1 in Z7
     
  6. Oct 29, 2006 #5
    Why does f(1)=0?
     
  7. Oct 29, 2006 #6

    matt grime

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    Because you said it did. You asked: why is the map f(x)=0 for all x not a homomorphism. Ans: because f(1) is not 1, contradicting the definition of ring homomorphism.
     
  8. Oct 29, 2006 #7
    Well the f(x)=0 wasn't part of the problem. It was just my own thinking. Does this still work? Somehow I'm still not getting where there is a contradiction in the definition.
     
  9. Oct 29, 2006 #8
    What I'm basically asking is, is there a ring homomorphism from Z5 to Z7. My book says no. Why is this?
     
  10. Oct 29, 2006 #9

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    If f(1)=1, then what is f(1+1), f(1+1+1), etc.? Eventually there will be a problem.
     
  11. Oct 29, 2006 #10

    matt grime

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    Does what still work?
     
  12. Oct 29, 2006 #11
    Yeah what's the problem?
    The identity requirement seems to hold. I'm really missing something. Could you spell it out for me?
     
  13. Oct 29, 2006 #12

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    So what are the elements 1, 1+1, 1+1+1, ... in Z5? Are any of them the same? If so, do they map to the same element in Z7, as they must?
     
  14. Oct 29, 2006 #13
    Are you saying that this function is injective and therefore not a ring homomorphism?
    Because I don't see how 3 in Z5 not being the same as 3 in Z7 is a reason for it not being a homomorphism.
     
  15. Oct 29, 2006 #14
    Can someone just update me on this?
     
  16. Oct 29, 2006 #15

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    Keep going. What is 5 in Z5? In Z7?
     
  17. Oct 30, 2006 #16
    OK. I think I got it. Thanks for the help.
     
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