Pure submodule explained

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

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 id_X ⊗ i : X ⊗ P → X ⊗ M (where the tensor products are taken over R) is injective.

Analogously, a short exact sequence

0 \longrightarrow A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \longrightarrow 0 of (left) R-modules is pure exact if the sequence stays exact when tensored with any (right) R-module X. This is equivalent to saying that f(A) 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 (a_ij) with entries in R, and any set y₁, ..., y_m of elements of P, if there exist elements x₁, ..., x_n in M such that

\sum ^{n} _{j =1} a _{ij} x _{j} = y _{i} for i =1 ,\ldots,m then there also exist elements x₁′, ..., x_n′ in P such that\sum ^{n} _{j =1} a _{ij} x' _{j} = y _{i} for i =1 ,\ldots,m

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

0 \longrightarrow A _{i} \longrightarrow B _{i} \longrightarrow C _{i} \longrightarrow 0 . ^[1]

Examples

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

Properties

Suppose

0 \longrightarrow A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \longrightarrow 0 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

0 \longrightarrow A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \longrightarrow 0 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.

References

1. For abelian groups, this is proved in