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Integral domain

  1. Dec 3, 2008 #1
    1. The problem statement, all variables and given/known data
    Given a,b,c in D, with a not 0, we have ab=ac implies b=c. Show that the commutative ring D is an integral domain.

    3. The attempt at a solution
    I don't know where to begin with this.
  2. jcsd
  3. Dec 3, 2008 #2
    You want to show that there are no 0 divisors. Look at what's given. Look at its contrapositive.
  4. Dec 3, 2008 #3
    So we want to assume it is not an integral domain, then show that b does not equal c?

    Well, I know it is possible for b to not equal c because if we are in say, Z mod 6, then [0]=[3]=[6]. But how do I generalize this? Is this the right method to go about it?
  5. Dec 3, 2008 #4
    You don't have to prove by contradiction. What I meant was that since we have

    a [tex]\ne[/tex] 0, ab = ac implies b = c,

    we also know

    a [tex]\ne[/tex] 0, b [tex]\ne[/tex] c implies ab [tex]\ne[/tex] ac. Letting b = 0 or c = 0 should get you what you want.
  6. Dec 3, 2008 #5
    I think Michael would be ashamed
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