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 exact sequence is a direct limit of split exact sequences.

Definition

Let R be a ring, let M be a 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 R-module X, the natural induced map idX ⊗ i : X ⊗ P → X ⊗ M 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 is a pure submodule of B.

Equivalent characterizations

Purity of a submodule 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 with entries in R, and any set y1,..., ym of elements of P, if there exist elements x1,..., xn in M such that
then there also exist elements x''1′,..., xn′ in ''P such that
Another characterization is: a sequence is pure exact if and only if it is the filtered colimit of split exact sequences

Examples

Properties

Suppose
is a short exact sequence of R-modules, then:
  1. C is a flat module if and only if the exact sequence is pure exact for every A and B. From this we can deduce that over a von Neumann regular ring, every submodule of every ''R-module is pure. This is because every module over a von Neumann regular ring is flat. The converse is also true.
  2. Suppose B'' is flat. Then the sequence is pure exact if and only if C is flat. From this one can deduce that pure submodules of flat modules are flat.
  3. Suppose C is flat. Then B is flat if and only if A is flat.
If 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''.