I am looking for a reference for the following result: Let $k$ be an algebraically closed field and $A$ be a finite dimensional $k$-algebra. If $A$ is basic, then every simple $A$-$A$ bimodule is 1-dimensional.
I believe this is correct but cannot find any reference that proves it. Does anyone know please?
PS, a $k$-algebra is basic if all simple (left or right) $A$-modules are $1$-dimensional.
Since $k$ is algebraically closed and $A$ is basic, $A$ is isomorphic to a quotient $kQ/I$ of a path algebra on a quiver $Q$ (which is, in fact, the Gabriel quiver of $A$) by an admissible ideal $I$ of $kQ$.
It follows easily from this the enveloping algebra $A\otimes A^{op}$ is the quotient of the path algebra on a certain quiver $Q\times Q$, which we can describe by calling it «the cartesian product of $Q$ with itself» by an ideal which is also not difficult to describe explicitly and which is clearly admissible. It follows from this that $A\otimes A^{op}$ is basic.
I don't know of any concrete reference, and people just use this fact — indeed, this is all very straightforward once you are familiar with Gabriel presentations of algebras.