I have been trying (without much success) to find a reference for learning about projective indecomposable $A$-$A$-bimodules: here $A$ denotes a finite-dimensional algebra over a field $K$.
I think I understand the picture when $A$ is basic and the underlying field is algebraically closed: using Lemma 5.3.8 and Corollary 5.3.10 in Representation theory: a homological algebra point of view, one concludes that the projective indecomposable bimodules must be of the form $Ae_i\otimes_K e_j A$, where $\{e_i\}_{i \in I}$ forms a complete irredundant set of primitive idempotents in $A$.
I would like to know what happens more generally, for instance, if $A$ is not necessarily basic and/or $K$ is not so nice.