= Center of a tensor product of associative algebras
{title2=$Z(A\otimes_kB)=Z(A)\otimes_kZ(B)$}
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 $a\otimes1$ 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>.
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