Exercise 11, § 12

Let A and B be K-algebras; suppose that the rings A and B are primitive and that their respective socles $\mathfrak{s}$ and $\mathfrak{t}$ (VIII, p. 130, Exercise 8) are not reduced to 0. Let D (resp. E) be the commutant of a minimal left ideal $\mathfrak{a}$ of A (resp. $\mathfrak{b}$ of B); suppose that the algebra $D\otimes E$ is simple. Prove that $A\otimes B$ is a primitive ring with socle $\mathfrak{s}\otimes \mathfrak{t}$ (observe that the $(A\otimes$ B)-module $\mathfrak{a}\otimes \mathfrak{b}$ is isotypical of finite length; use Exercise 9, a) of VIII, p. 131).