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.
The problem is simple: if $A$ and $B$ are both left $R$-modules, we'd want to equate $(ra,b)$ with $(a,rb)$ in the tensor product. But then we'd have $(rsa,b)=(a,rsb)$ and $(rsa,b)=(sa,rb)=(a,srb)$! So this construction forces $R$ to act commutatively on $A\otimes B$.
In general the tensor product of a right and a left $R$-module is merely an abelian group. There's no module structure for essentially the same reason as above: for instance the natural left module structure would come from $B$, but we'd be forced again to say $$(a,rsb)=r(a,sb)=r(as,b)=(as,rb)=(a,srb)$$