A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms short exact sequences into short exact sequences. A covariant functor is called left exact if it preserves the exactness of all sequences 0⟶A⟶B⟶C, and it is called right exact if it preserves the exactness of all sequences A⟶B⟶C⟶0. ("Left" and "right" are interchanged in the corresponding definitions for contravariant functors.) A functor is exact iff it is both left and right exact.

flat module | hom | injective module | projective module | short exact sequence