# Math Help - Units

1. ## Units

Given a ring $R$ and $r,s \in R$, show $1+rs$ is a unit $\Longleftrightarrow 1+sr$ is a unit.

2. Originally Posted by mathman88

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.