In mathematics, specifically

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

, the Eilenberg–Watts theorem tells when a

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

between the categories of modules is given by an application of a

tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

. Precisely, it says that a functor

F : \mathbf_R \to \mathbf_S

is additive, is right-exact and preserves coproducts if and only if it is of the form

F \simeq - \otimes_R F(R)

. For a proof, se
The theorems of Eilenberg & Watts (Part 1)

References

*Charles E. Watts, Intrinsic characterizations of some additive functors, Proc. Amer. Math. Soc. 11, 1960, 5–8. *Samuel Eilenberg, Abstract description of some basic functors, J. Indian Math. Soc. (N.S.) 24, 1960, 231–234 (1961).

References

Further reading