For $x \in Z_a$, $y \in Z_b$, what is $x \otimes y \in Z_a \otimes Z_b$?

It can't be identified with a bilinear machine into $Z$, but it can be identified with a homomorphism from $Z_a$ into $Z_b$: the function which sends $t \in Z_a$ to $t * x * y$ mod $b$.