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.