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Help with abstract algebra proof

  1. Feb 10, 2010 #1
    prove that a finite ring with identity has characteristic n for some n>0.
    been trying for a while getting nowhere any ideas?
     
  2. jcsd
  3. Feb 10, 2010 #2

    Dick

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    What have you been trying? The ring is FINITE. If f(n)=1+1+1+... n times, you can't get different values of f(n) for all n.
     
  4. Feb 10, 2010 #3
    you are assuming that we are dealing with numbers and with our normal addition. addition can be defined in any way. thats my problem just becuase it is a ring does not mean we are dealing with numbers or at the least the regular def of addition and multiplication
     
  5. Feb 10, 2010 #4

    Dick

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    Your ring has an identity. I called that '1'. Your ring also has an addition operation, I called that '+'. Why do you think I'm assuming anything?
     
  6. Feb 10, 2010 #5
    how do u know what happens when u add to numbers, how do u know that they get larger define larger. we may not be dealing with numbers all we know is that its a ring. we dont know what 1 is it is just the muliplictive inverse. it may not be a number
     
  7. Feb 10, 2010 #6

    Dick

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    Who said that anything was larger? I just said the f(n) couldn't ALL be different because your ring is finite. Besides 1 is a multiplicative identity, not an inverse. I really don't think you are in a mood to listen to any advice here.
     
  8. Feb 10, 2010 #7
    im in a good mood sorry if u get affended i really appericate the help but i dissagree i a few things
     
  9. Feb 10, 2010 #8
    how to u do know that 1+1+1...n times isnt just one? we dont know how addition is defined?
     
  10. Feb 11, 2010 #9

    Dick

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    Ok, sorry to be thin skinned, but you objecting to things I'm not even saying while not paying any attention to my original suggestion. Once more, your ring R is FINITE. That means f(n)=f(m) for some n and m. See post 2 for the definition of f(n). What does that tell you about characteristic? If you want to object to more things that's ok, because I'm not here. I'm going to bed.
     
  11. Feb 11, 2010 #10

    Dick

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    If 1+1+... n times=1, then 1+1+.. n-1 times=0. We do know that much about how addition is defined. Characteristic!
     
  12. Jul 19, 2011 #11
    Since the ring is finite, we know that there exist some integers n, and m, such that n*1 is equal to m*1, and let n>m. (this was already suggested) Try taking the difference of these to find if there is a characteristic.
     
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