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Ring proof

  1. Dec 13, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that Z with the following addition and subtraction is a ring.

    2. Relevant equations
    a[tex]\oplus[/tex]b = a + b - 1 and a[tex]\odot[/tex]b = ab - (a + b) + 2

    3. The attempt at a solution I proved all the axioms for addition. I'm stuck on the multiplication part.

    (a[tex]\odot[/tex]b)[tex]\odot[/tex]c = (ab-(a+b)+2)c - (ab-(a+b)+2+c) + 2

    a[tex]\odot[/tex](b[tex]\odot[/tex]c) = a(bc-(b+c)+2) - (a+bc-(b+c)+2) + 2

    How are these equal? I know it's a ring because a couple problems later, my books wants me to prove that it's an integral domain...
  2. jcsd
  3. Dec 13, 2009 #2


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    I think they are equal. Just expand them out.
  4. Dec 13, 2009 #3
    (ab-(a+b)+2)c - (ab-(a+b)+2+c) + 2 =

    abc-ac-bc+2c-ab+a+b-2-c+2 = abc-ac-bc-ab+c+a+b

    a(bc-(b+c)+2) - (a+bc-(b+c)+2) + 2 =

    abc-ab-ac+2a-a-bc+b+c-2+2 = abc-ab-ac-bc+a+b+c

    It looks so much clearer now. My own handwriting was deceiving me... Gah, that's the second stupid question I've posted this weekend...
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