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Thread: Need help with invertibility proof

  1. #1
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    Need help with invertibility proof

    Hello.

    I would use some help with following task:
    Prove that when in ring R products xy and yx are invertible then elements x and y are also invertible.

    Thanks in advance.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Need help with invertibility proof

    What does it mean by definition that xy is invertible?

    If you have the definition then you can use the fact that the multiplication is an associative operation.
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    Re: Need help with invertibility proof

    It means that there exists (xy)^-1 so that (xy)^-1 * xy = 1
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    MHF Contributor Siron's Avatar
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    Re: Need help with invertibility proof

    Quote Originally Posted by rain1 View Post
    It means that there exists (xy)^-1 so that (xy)^-1 * xy = 1
    Yes, xy is invertible is there exists an unique z \in R such that (xy)z=1.
    Now, use the associtativity.
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  5. #5
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    Re: Need help with invertibility proof

    You mean like this?  (xy)^{-1}\cdot (xy) = x\cdot (y^{-1}\cdot y)\cdot x^{-1} = x\cdot 1x^{-1} = 1 \cdot 1 =1? Thats all?
    Last edited by rain1; Mar 24th 2013 at 12:38 PM.
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  6. #6
    MHF Contributor Siron's Avatar
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    Re: Need help with invertibility proof

    Quote Originally Posted by rain1 View Post
    You mean like this?  (xy)^{-1}\cdot (xy) = x\cdot (y^{-1}\cdot y)\cdot x^{-1} = x\cdot 1x^{-1} = 1 \cdot 1 =1? Thats all?
    In fact, what I mean is the following.
    Suppose xy is invertible then there exists an unique z \in R such that (xy)z=1. Since 1=(xy)z=x(yz) (here I use the associativity), we find that x is invertible with yz as the inverse element.
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