Pure submodule

In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave short exact sequences exact after tensoring, a pure submodule defines a short exact sequence that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure submodule defines a short exact sequence which is a direct limit of split exact sequences, each defined by a direct summand.
Definition.
Let "R" be a ring (associative, with 1), and let "M" and "P" be modules over "R". If "i": "P" → "M" is injective then "P" is a pure submodule of "M" if, for any "R"-module "X", the natural induced map on tensor products "i" ⊗ id"X" : "P" ⊗ "X" → "M" ⊗ "X" is injective.
Analogously, a short exact sequence
of "R"-modules is pure exact if the sequence stays exact when tensored with any "R"-module "X". This is equivalent to saying that "f"("A") is a pure submodule of "B".
Purity can also be expressed element-wise; it is really a statement about the solvability of certain systems of linear equations. Specifically, "P" is pure in "M" if and only if the following condition holds: for any "m"-by-"n" matrix ("a""ij") with entries in "R", and any set "y"1, ..., "y""m" of elements of "P", if there exist elements "x"1, ..., "x""n" in "M" such that
then there also exist elements "x"1′, ..., "x""n"′ in "P" such that
Examples.
is a short exact sequence of "R"-modules, then: 
Equivalent characterization.
A sequence is pure exact if and only if it is the filtered colimit (also known as direct limit) of split exact sequences