Given a quiver, we know that it is easy to get the indecomposable projective modules, but the indecomposable injective modules are not easy to get.
How do you get the indecomposable injective modules from indecomposable projective modules?
For example, $Q = (Q_0,Q_1)$ is the quiver $\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} 4 & \ra{\alpha_{43}} & 3 \\ \da{\alpha_{42}} & & \da{\alpha_{31}} \\ 2 & \ras{\alpha_{21}} & 1 \\ \end{array}$ and $\mathcal{I} = \langle\alpha_{42}\alpha_{21}-\alpha_{43}\alpha_{31}\rangle$ the admissible ideal of$ KQ$.