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 (known as a pure exact sequence) that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences.

Definition

be a ring be a (left) module be a submodule be the natural injective is a is injective.

Analogously, a short exact sequence

-modules is pure exact.

Equivalent characterizations

matrix such that

such that

Another characterization is: a sequence is pure exact if and only if it is the filtered colimit (also known as direct limit) of split exact sequences

Examples

• Every direct summand of M is pure in M. Consequently, every subspace of a vector space over a field is pure.

Properties

-modules, then:

1. is a flat module. From this we can deduce that over a von Neumann regular ring, every submodule of every-module is pure. This is because every module over a von Neumann regular ring is flat. The converse is also true.

2. is flat. From this one can deduce that pure submodules of flat modules are flat.

3. is flat.

is a finitely presented.

References