Indecomposable Ideals in Incidence Algebras

The elements of a finite partial order $P$ can be identified with the maximal indecomposable two-sided ideals of its incidence algebra $\text{A}$, and then for two such ideals, $I\prec J⟺IJ≠0$. This offers one way to recover a poset from its incidence algebra. In the course of proving the above, we classify all of the two-sided ideals of $\text{A}$.