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Math Help - One Sided Identities

  1. #1
    Junior Member
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    Mar 2010
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    One Sided Identities

    I'm having trouble with this problem. Here is all the problem states:
    Prove: Let <A, O> be a system (where O is the operator) with a left identity of g and a right identity of d. (So for all elements of A, a, aOd = a and gOa = a)
    Show that g = d.

    As corollaries, show that a two-sided identity is unique, and if O is commutative, then <A,O> has at most one left - identity.

    I understand what they are asking for but I can't figure out how to prove it without assuming O is commutative. Thanks for the help in advance.
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  2. #2
    Senior Member
    Joined
    Feb 2008
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    Quote Originally Posted by jameselmore91 View Post
    I'm having trouble with this problem. Here is all the problem states:
    Prove: Let <A, O> be a system (where O is the operator) with a left identity of g and a right identity of d. (So for all elements of A, a, aOd = a and gOa = a)
    Show that g = d.

    As corollaries, show that a two-sided identity is unique, and if O is commutative, then <A,O> has at most one left - identity.

    I understand what they are asking for but I can't figure out how to prove it without assuming O is commutative. Thanks for the help in advance.
    gOa=a\;\forall\;a\in A \implies gOd=d.

    aOd=a\;\forall\;a\in A \implies gOd=g.

    The conclusion follows.
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