# Special multiserial algebra and monomial special multiserial algebra

Let $K$ be an algebraically closed field. $Q=(Q_0,Q_1)$ is a quiver($Q_0$ is a finite set of vertices and $Q_1$ is a finite set of arrows). An algebra $KQ/I$ is called monomial if $I$ is a ideal consisting of monomial relations. A finite dimensional algebra $A$ is special multiserial if it is Morita equivalent to an algebra $KQ/I$ such that for all $a \in Q_1$ there exists at most one arrow $b \in Q_1$ such that $ab \not \in I$ and there exists at most one arrow $c \in Q_1$ such that $ca\not \in I$.

Now assume that $A$ is special multiserial. I have seen in a place that "by successively factoring out the socles of indecomposable projective-injective modules, we get a monomial special multiserial algebra $B$. I don't know why the above conclusion hold. Also I want to know whether it is hold for general quiver algebras. Thank you for some reference about it and help.

This is not true as stated. For example take the path algebra of the quiver with 5 vertices and 5 arrows: $1\to 2\to 4$, $1\to 3\to 4$, and $1\to 5$. Impose the relation $(1\to 2\to 4)=(1\to 3\to 4)$. Then, this algebra is obviously special multiserial, but it does not have indecomposable projective-injective modules.
An easier example in the general case is the path algebra of $D_4$ with non-special-multiserial orientation. It also does not have indecomposable projective-injective modules.