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Math Help - Algebra, Problems For Fun (6)

  1. #1
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    Algebra, Problems For Fun (6)

    Let R be a unitary ring. Suppose that ab=0 \Longrightarrow ba=0, for any a,b \in R. Prove that ab=1 \Longrightarrow ba=1, for any a,b \in R.
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    So... in an integral domain, "one-sided invertible \Leftrightarrow invertible" ?
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    Quote Originally Posted by clic-clac View Post
    So... in an integral domain, "one-sided invertible \Leftrightarrow invertible" ?
    that's right!
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    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by NonCommAlg View Post
    Let R be a unitary ring. Suppose that ab=0 \Longrightarrow ba=0, for any a,b \in R. Prove that ab=1 \Longrightarrow ba=1, for any a,b \in R.
    Hi NonCommAlg.

    _____ ab=1

    \implies\ ab-1=0

    \implies\ b(ab-1)=b0=0=0b=(ab-1)b

    \implies\ bab-b=ab^2-b

    \implies\ bab=ab^2

    \implies\ bab-ab^2=0

    \implies\ (ba-ab)b=0

    \implies\ b(ba-ab)=0

    \implies\ ab(ba-ab)=a0=0

    \implies\ 1(ba-ab)=0

    \implies\ ba-ab=0

    \implies\ ba=ab=1
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  5. #5
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    Quote Originally Posted by TheAbstractionist View Post
    Hi NonCommAlg.

    _____ ab=1

    \implies\ ab-1=0

    \implies\ b(ab-1)=b0=0=0b=(ab-1)b

    \implies\ bab-b=ab^2-b

    \implies\ bab=ab^2

    \implies\ bab-ab^2=0

    \implies\ (ba-ab)b=0

    \implies\ b(ba-ab)=0

    \implies\ ab(ba-ab)=a0=0

    \implies\ 1(ba-ab)=0

    \implies\ ba-ab=0

    \implies\ ba=ab=1
    that's correct. good work! an almost similar way: 0=b(1-ab)=b-bab=(1-ba)b=b(1-ba). therefore 1-ba=ab(1-ba)=0.
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