Let $R$ be a ring with unity different from $0$, and it can be non-commutative.
Let $I=Re$ be a minimal ideal of $R$ generated by an idempotent $e$.
Q. If $Re=Re'$ for some idempotent $e'$ of $R$ then is it necessary that $e=e'$?
If the answer is NO in general, assuming $R$ to be semisimple ring, is the answer YES?
Even in semisimple rings the answer is “no.”
$R\begin{bmatrix}0&1\\0&1\end{bmatrix}=R \begin{bmatrix}0&0\\0&1\end{bmatrix} $ in $M_2(\mathbb Q)=R$, for example.