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Math Help - Units

  1. #1
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    Units

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

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

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

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

    a similar argument proves the converse of the claim.
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