A question about tensor product and multiplication?

Question

Let $k$ be a commutative ring, $A$ be an algebra over $k$. The tensor product $A\otimes A$ is over $k$. If $\sum_{i} a_i \otimes b_i=\sum_j c_j\otimes d_j$, where $a_i,b_i,c_j,d_j\in A$, I wonder if $\sum_i a_i b_i=\sum_j c_j d_j$? Or if that holds under some conditions?

Thank you!

Answer 1

(This works only for commutative $k$-algebras.)

Hint. One can define a ring homomorphism $A\otimes_kA\to A$ by $\sum a_i\otimes b_i\mapsto\sum a_ib_i$.