I am stuck on proving a left ideal $I$ of a ring $R$ is a direct summand of $R$ if and only if $I = Rr$ with $r^2 = r$. Could you help me with that? Any help will be very appreciated. Thanks!
$R(1-r)$ is your candidate for a complement
In the other direction, if $1=a+b$ where $a\in I$ and $b\in J$ and $I\oplus J=R$, $a$ is your candidate for $r$.