Tensor product of associative algebras ($(a\otimes b)(c\otimes d)=ac\otimes bd$)

For two unital associative algebras over a field $k$, their tensor product carries the multiplication in the title, extended by bilinearity, with identity $1\otimes 1$. Choosing a basis $(b_j)$ of the second factor gives the basis of a module $(1\otimes b_j)$ for the resulting left $A$-module; dual basis functionals recover the coefficients uniquely.

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.