Proposition 2

Let S be a simple module over the ring $A = A_1\otimes A_2$, finite-dimensional over K. For $i\in \{1,2\}$, there exists a simple $A_i$-module $S_i$ such that the $A_i$-module S is isotypical of type $S_i$. The A-module S is isomorphic to a quotient of the A-module $S_1\otimes S_2$.

Since S is a nonzero $A_1$-module that is finite-dimensional over K, it has finite length over $A_1$ and there exist a simple left $A_1$-module $S_1$ and a nonzero $A_1$-linear mapping from $S_1$ to S. We endow $M_2=$ Hom$_{A_1}(S_1,S)$ with a left $A_2$-module structure defined by the external law $(a_2, u)\mapsto (a_2)_S\circ u$. We have $M_2\not= 0$ by construction, and $M_2$ is finite-dimensional over K. We can therefore find a simple left $A_2$-module $S_2$ and a nonzero $A_2$-linear mapping $\varphi : S_2\rightarrow M_2$. We define a nonzero A-linear mapping $\psi$ from $S_1\otimes S_2$ to S by

for every $s_1\in S_1$ and $s_2\in S_2$. Since S is a simple A-module and $\psi$ is not zero, $\psi$ is surjective and S is isomorphic to a quotient of $S_1\otimes S_2$. For $i\in \{1,2\}$, the $A_i$-module $S_1\otimes S_2$ is isotypical of type $S_i$, and therefore so is the $A_i$-module S (VIII, p. 61, Proposition 2).

For any K-algebra B, we denote by $\mathscr{S}_K(B)$ the set of classes of simple (VIII, p. 51) left B-modules that are finite-dimensional over K.