A module having dual properties with respect to a free module, as enumerated below.
1. Every free module is projective; every cofree module is injective.
2. For every module , there is a surjective homomorphism from a free module to ; for every module , there is an injective homomorphism from to a cofree module.
3. A module is projective iff it can be completed by a direct sum to a free module; a module is injective iff it can be completed by a direct product to a cofree module.
Every cofree module over a unit ring is isomorphic to a direct product
indexed on some set .