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Invertible elements in a commutative ring with no zero divisors

  1. Aug 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Suppose that a commutative ring R, with a unit, has no zero divisors. Does that necessarily imply that every nonzero element of R is invertible?


    2. Relevant equations



    3. The attempt at a solution

    We have to show that there exists some b in R such that ab = e. Having no zero divisors implies that if b[itex]\neq[/itex]0 then ab[itex]\neq[/itex]0.

    To show that every nonzero element of R is not invertible we must find a case where ab = c for some c in R and c [itex]\neq[/itex] e.

    The question seems easy but I can't wrap my head around how to write it down.
     
  2. jcsd
  3. Aug 12, 2013 #2
    Can you think of a particular example of such a ring? Is every non-zero element in it invertible?
     
  4. Aug 12, 2013 #3
    Oh can we just give an example, the integers form such a ring but the only invertible elements are 1 and -1. Therefore every non-zero element is not invertible and the question false.

    Thanks!
     
  5. Aug 12, 2013 #4
    A counter-example is a perfectly good way to show that some particular statement is not true.
     
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