I was reading something about tensor product (balanced product) of modules (over an arbitrary ring $R$), but I cannot realize why we need a left and a right module. Would it be the same with two right (or left) modules? What make this situation so special?
Edit: Another question
Is the tensor product also a module over R or it just a left (right) one? Why?
The definition I've read is the one that uses factor groups of free abelian groups.