I've managed to get a weak grip on what the tensor product of two vector spaces is. I'm now trying to understand the tensor product of two algebras.
I understand that we define $(v_1\otimes w_1)(v_2\otimes w_2):=(v_1v_2)\otimes (w_1w_2)$. In order for this to be bilinear we must have relations like $x(y+z)=xy+xz$ (x,y,z are tensors here). I can't show that this is true: you can't in general simplify two tensors $y$ and $z$ into a new tensor of the form $v\otimes w$ (i.e the sum won't necessarily be a pure tensor). But then the multiplication isn't defined for a sum of pure tensors, only for pure tensors. What can I do here? Thanks for any replies!