Suppose $R$ is a ring (possibly noncommutative), $I$ is a minimal left ideal in it, and $I^2\neq 0$, show that $I=Re$ for some idemopotent $e$.
It is easy to show that we can find some $x\in I$, such that $I=Ix$, so $I=Rx=Rx^n=Ix^n$, for all $n\geq 1$, but how to construct the idemopotent $e$ using $x$, I guess I must have failed to realize some key point.