Definition 9

A submodule $M$ of an $A$-module $E$ is called a direct factor of $E$ if it has a supplementary submodule in $E$.

When this is so, $E/M$ is isomorphic to a supplement of $M$ (Proposition 13).

A submodule does not necessarily admit a supplement (Exercise 11). When a submodule is a direct factor, it has in general several distinct supplements; these supplements are however canonically isomorphic to one another (Corollary to Proposition 13).