Let $A$ be a noncommutative $\mathbb Q$-algebra, and let $a\in A$. If we know that $a$ has a left inverse in $A\otimes \mathbb Q_\ell$ for a prime $\ell$, then we can conclude that $a$ is already left invertible in $A$.
Now suppose instead that the equation $Xa\pi =\pi$ has a solution in $A\otimes \mathbb Q_\ell$, where $\pi$ is an idempotent element of $A$. Does it follow that it is already solvable in $A$?