Does affine rank 3 acyclic quiver $\tilde{A}_{12}$ with arrows $2 \to 1$, $3\to 2$ and $3 \to 1$ has admissible ideal?
For any finite quiver $Q$, a two-sided ideal $I$ of $KQ$ is said to be admissible if there exists $m \geq2$ such that $R^m_Q\subseteq I\subseteq R^2_Q$.
So $\langle 3\xrightarrow[]{\beta}2 \xrightarrow[]{\alpha}1 - 3\xrightarrow[]{\gamma} 1\rangle$ cannot be an admissible ideal. Am I right?
Can $\langle 3\xrightarrow[]{\beta}2 \xrightarrow[]{\alpha}1$ be an admissible ideal?
If not? So how can we find its indecomposable projective and injective modules?
I appretiate if someone gives me some ideas.
Thanks!