I am reading these online notes(http://people.virginia.edu/~mve2x/7752_Spring2010/lecture5.pdf), and on the bottom of page 2 the following exercise occurs.
Define tensor multiplication of two $R$-algebras $A$ and $B$ as:
$(a_1 \otimes b_1) \cdot (a_2 \otimes b_2) = a_1 a_2 \otimes b_1 b_2$.
Prove this is bilinear.
The first property of a function being bilinear is $(u+w) \cdot v = (u \cdot v) + (w \cdot v)$, so I tested the following:
$(a_1 \otimes b_1 + a_3 \otimes b_3) \cdot (a_2 \otimes b_2) = ???$
Tensor addition is not defined when one of the elements differs, so I am unsure of where to go beyond this.