I'm trying to figure out why an element $u$ in some ring is invertible with inverse $z$ if any only if
- $uzu=u$ and $zu^2z=1$
OR
- $uzu=u$ and $z$ is the unique element meeting this condition.
Clearly, both conditions follow if $u$ is a unit with inverse $z$. However, I can't see why either condition implies that $z=u^{-1}$.
I haven't been able to make any decent progress on my own, so does anyone have hints or suggestions on where to go? Thanks.
Edit: From Qiaochu's hint, $zu$ and $uz$ are idempotent. So $(zu)^2=zu$. But $zu$ has right inverse $uz$, so $(zu)^2(uz)=(zu)(uz)\implies zu=1$. The analogous argument for $uz$ shows $uz=1$, so $z=u^{-1}$.
Does anyone have an idea for the second?