I has a question about indecomposable modules over monomial algebras.
An admissible ideal $I$ of a path algebra $kQ$ is called monomial if it is generated by some paths of length at least two. The quotient algebra $A = {\bf k}Q/I$ is called a monomial algebra.
Let $A={\bf k}Q/I$ be a monomial algebra. We call $A$ \textit{acyclic} if $Q$ has no cycles, and \textit{cyclic} otherwise. Recall that the set $\{e_{i}\mid i\in{Q_{0}}\}$ is a complete set of primitive orthogonal idempotents of $A$. For each $i\in Q_{0}$, we denote by $S(i)=e_{i}A/{\rm rad}\,e_{i}A$, $P(i)=e_{i}A$ and $I(i)=D(A e_{i})$ the simple module, indecomposable projective module, and indecomposable injective module at $i$, respectively, where $D={\rm Hom}_{\bf k}\,(-,{\bf k})$ denotes the standard duality.
Let $X$ be an $A$-module with $top X\cong S(j_{1})\oplus S(j_{1})$ and $soc X\cong S(i_{1})\oplus S(i_{1})\oplus S(i_{2})\oplus S(i_{2})\oplus S(i_{3})\oplus S(i_{3})$, where $j_{1},i_{1},i_{2},i_{3}\in Q_{0}$. Suppose that $I(i_{1})$, $I(i_{1})$, and $I(i_{1})$ are uniserial and projective-injective modules. Then $X$ is decomposable. I attempted to prove this result but failed, and I haven’t found a counterexample either. Could you help by providing a proof idea or a counterexample? Thank you
