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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$.

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    Aimin: xymatrix [does not](http://meta.math.stackexchange.com/questions/3984/when-will-math-se-support-the-beginxymatrix-environment) work in MathJax. You can either make a picture on your computer using dvipng or some similar utility and upload it here, or you can try to use some kind of [workaround](http://meta.math.stackexchange.com/questions/2324/how-to-draw-a-commutative-diagram) using the features that are supported in MathJax.2012-11-07

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