Let $A_1, A_2$ denote two unital, commutative $k$ algebras, where $k$ is a field. Then by definition $A_i$ are rings with a bilinear operation on $k$.
When we speak of the tensor product $A_1 \otimes A_2$, this is also a unital commutative ring.
I want to make sure I understand correctly; is the additive abelian group of $A_1 \otimes A_2$ just $\{ 0\}$? How do we define addition on this ring?
If $a_1 \otimes a_2, b_1 \otimes b_2 \in A_1 \otimes A_2$ and their addition would defined as $a_1 \otimes a_2 + b_1 \otimes b_2 = a_1 + b_1 \otimes a_2 + b_2$, then $a_1 \otimes a_2 = a_1 \otimes 0 + 0 \times a_2 = 0$, no?
How do we define addition in this ring, and do I misunderstand something more basic?