The author extends the Azumaya\mhy Nakayama theorem concerning the Kronecker product of irreducible algebras of linear transformations [cf.\ \n N. Jacobson\en, {\it Structure of rings}, Amer. Math. Soc., Providence, R.I., 1964; MR0222106 (36 \#5158)]. Applying the extended theorem he obtains the following results. Let $P_i,\ i=1,2$, be primitive algebras (over the same field $F$) with nonzero socles $S(P_i)$ and let $D_i$ be the associated division algebras. Then: (i) if the algebra $D_1\otimes D_2$ is simple and $S(D_1\otimes D_2)\not=0$ then the algebra $P_1\otimes P_2$ is primitive and $S(P_1\otimes P_2)\not=0$; (ii) if $P_1\otimes P_2$ is primitive and $S(P_1\otimes P_2)\not=0$ then $D_1\otimes D_2$ is primitive and $S(D_1\otimes D_2)\not=0$;\break (iii) $D_1\otimes D_2$ is simple if and only if $S(P_1)\otimes S(P_2)$ is simple. \par \{Reviewer's remark: Some of the quoted results were also obtained by \n W. K. Nicholson\en and \n J. F. Watters\en [Comm. Algebra {\bf 9} (1981), no. 3, 299--311; MR0603351 (82h:16002)].\}