Looking for citable reference for a well-known fact about tensor product of finite dimensional algebras over an algebraically closed field

Question

Let $K$ be an algebraically closed field and let $A$ and $B$ be finite dimensional algebras over $K$. Let $e_1,\ldots, e_n$ be orthogonal primitive idempotents of $A$ summing to $1$ and $f_1,\ldots, f_m$ be orthogonal primitive idempotents of $B$ summing to $1$. Then the $e_i\otimes f_j$ form a set of orthogonal primitive idempotents of $A\otimes_K B$ summing to $1$. Moreover, $(A\otimes_K B)(e_i\otimes f_j)\cong (A\otimes_K B)(e_{i'}\otimes f_{j'})$ if and only if $Ae_i\cong Ae_{i'}$ and $Bf_j\cong Bf_{j'}$. Note that this need not hold if $K$ is not algebraically closed.

I know how to prove these things but I want to cite this in a paper I'm writing and couldn't find this spelled out in any of my usual references on finite dimensional algebras. I don't really want to put a proof in my paper as it is a bit far a field. Actually, what I want to use this for is to conclude that the basic algebra of $A\otimes_K B$ is the tensor product of the basic algebras of $A$ and $B$ (again with $K$ algebraically closed).

I would greatly appreciate any reference, particularly, to a book.

Answer 1

A treatment of this can be found in chapter IV.11. in the book "Frobenius algebras I" by Skowronski and Yamagata with proofs in the case $A=B^{op}$. But the proofs work exactly the same in the general case with the exact same proofs so it can be used as a reference for the general case I think.