Tensor product

The tensor product is a bifunctor in the category of modules over a fixed ring $R$. In the subcategory of algebras over $R$, the tensor product is just the cofibered product over $R$.

Definition

The tensor product of two $R$-modules $M$ and $M'$, denoted by $M\otimes _{R}M'$, is an $R$-module $T$ satisfying the universal property

Functoriality

The functor $-\otimes _{R}M$ is right-exact from the category of (right) $R-modules$ to the category of $R$-modules.  

The derived functors $Tor_{R}^{n}(-,-)$.