Lemma 2

Let I and J be finite sets, $k$ be a commutative ring, and A and B be $k$-algebras. Denote by $\mathbf{M}_I(A)$ the $k$-algebra of square matrices of type $(I,I)$ with entries in A, and define the $k$-algebras $\mathbf{M}_J(B)$ and $\mathbf{M}_{I\times J}(A\otimes_KB)$ likewise. There exists a unique $k$-algebra isomorphism

such that $\varphi ((a_{ii'})\otimes (b_{jj'}))$ is the matrix with entry of index $((i, j),(i', j'))$ equal to $a_{ii'}\otimes b_{jj'}$.

The existence of a $k$-linear bijection $\varphi$ with the property stated in the lemma follows from the compatibility of the tensor product with direct sums (II, §3, No. 7, p. 255, Proposition 7). The fact that $\varphi$ is an algebra homomorphism follows from the definition of a product matrix.