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Thread: Units

  1. #1
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    Units

    Given a ring $\displaystyle R $ and $\displaystyle r,s \in R $, show $\displaystyle 1+rs $ is a unit $\displaystyle \Longleftrightarrow 1+sr $ is a unit.
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  2. #2
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    Quote Originally Posted by mathman88 View Post

    Given a ring $\displaystyle R $ and $\displaystyle r,s \in R $, show $\displaystyle 1+rs $ is a unit $\displaystyle \Longleftrightarrow 1+sr $ is a unit.
    this is a tricky question! suppose $\displaystyle 1+rs$ is a unit and $\displaystyle t$ is its inverse, then we have:

    $\displaystyle 1=1+sr-sr=1+sr-s(1+rs)tr=1+sr-str-srstr$

    $\displaystyle =1-str + sr(1-str)=(1+sr)(1-str).$ so we proved that $\displaystyle (1+sr)(1-str)=1.$

    it's easy now to show that $\displaystyle (1-str)(1+sr)=1.$ thus $\displaystyle 1-str$ is the inverse of $\displaystyle 1+sr.$

    a similar argument proves the converse of the claim.
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