Center of a tensor product of associative algebras ($Z(A{\otimes }_{k}B)=Z(A){\otimes }_{k}Z(B)$)

For associative algebras of finite dimension over a field $k$, write a central tensor as $x={\sum }_j{a}_j\otimes b_j$ in a basis of $B$. Commutation with every $a\otimes 1$ forces each $a_j$ into the center of an associative algebra $Z(A)$. Reexpand in a basis of $Z(A)$ and commute with every $1\otimes b$ to force the second coefficients into $Z(B)$. The reverse inclusion follows by multiplying pure tensors.