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Homework Help: Left and right Identities

  1. Jan 16, 2012 #1
    If an operation has two left identities, show that it has no right identity.
    Let e[itex]_{1}[/itex] and e[itex]_{2}[/itex] be left identities such that e[itex]_{1}[/itex]≠e[itex]_{2}[/itex]. Assume there exist a right identity and call it r.

    Then we have that
    e[itex]_{2}[/itex]x=x and

    From here I want to try and show that there can not be a right identity but I don't see where to go.
  2. jcsd
  3. Jan 16, 2012 #2


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    This is fine so far.

    Try evaluating e1r. What two pieces of information can you conclude? Similarly...
  4. Jan 16, 2012 #3
    wouldnt I get



    So we get e[itex]_{2}[/itex] and e[itex]_{1}[/itex] are equal contradicting that they were distinct.

    Is that right?
  5. Jan 16, 2012 #4


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  6. Jan 16, 2012 #5
    thanks, I was trying to figure it out with the x's and I couldnt come to any contradiction
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