Quivers without loops admit global dimension 2

Let $Q$ be a finite quiver without loops. Then there is an admissible ideal $I$ such that the algebra $kQ\mathrm{/}I$ has global dimension at most two and is (strongly) quasi-hereditary. In addition some other (strongly) quasi-hereditary algebras $kQ\mathrm{/}{I}^{\mathrm{\prime }}$ are constructed with bigger global dimension.