Corollary 1

The module $S_1\otimes S_2$ over the ring $A_1\otimes A_2$ is semisimple (resp. isotypical, simple) if and only if the ring $D = D_1\otimes D_2$ is semisimple (resp. simple, a field). In particular, $S_1\otimes S_2$ is simple if the commutant of $S_1$ or $S_2$ is equal to K.

By Theorem 2, the module $S_1\otimes S_2$ over the ring $D_1\otimes D_2$ is semisimple (resp. isotypical, simple) if and only if the right D-module $(D_1\otimes D_2)_d$ is (VIII, p. 109, Proposition 10). Now the right D-module $D_d$ is simple if and and only if D is a field; it is isotypical (resp. semisimple) if and only if the ring D is simple (resp. semisimple) (VIII, p. 120, Definition 1; VIII, p. 121, Corollary 1; and VIII, p. 137, Proposition 2).