Pure submodule

Similarly a flat module is a direct limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences.

Let R be a ring (associative, with 1), let M be a (left) module over R, let P be a submodule of M and let i: P → M be the natural injective map.

Then P is a pure submodule of M if, for any (right) R-module X, the natural induced map idX ⊗ i : X ⊗ P → X ⊗ M (where the tensor products are taken over R) is injective.

Purity of a submodule can also be expressed element-wise; it is really a statement about the solvability of certain systems of linear equations.

is pure-exact, and F is a finitely presented R-module, then every homomorphism from F to C can be lifted to B, i.e. to every u : F → C there exists v : F → B such that gv=u.