A question about the socle of an algebra.

Question

Let $A$ be a finite dimensional algebra over a field K. Suppose $e$ is an idempotent such that $Ae$ is the direct sum of all indecomposable injective-projective left $A$-modules. Is that $soc(_AA) \in add(soc(Ae))$ holds?

Answer 1

No, this is not true. For example, an algebra need not have any projective-injective modules. Consider the path algebra of the quiver $1\to 2\leftarrow 3$. Then none of the projective modules is injective, hence $e=0$, but $soc(A)=3S_2$.

It is true for algebras of positive dominant dimension, i.e. if the algebra, considered as a left module over itself embeds into a projective-injective module. Then, since $\operatorname{soc}$ is a left exact functor, $\operatorname{soc}(A)\in\operatorname{add} \operatorname{soc}(I)\subseteq \operatorname{add}\operatorname{soc}(Ae)$ for this projective-injective module $I$. This class of algebras for example contains Auslander algebras as well as selfinjective algebras.