# 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 (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 y$_1$, ..., y$_m$ of elements of P, if there exist elements x$_1$, ..., x$_n$ in M such that
    $\sum _{j=1}^{n}a_{ij}x_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m$
then there also exist elements x$_1$′, ..., x$_n$′ in P such that
    $\sum _{j=1}^{n}a_{ij}x'_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m$
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
    $0\longrightarrow A_{i}\longrightarrow B_{i}\longrightarrow C_{i}\longrightarrow 0.$

## 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:
* 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.
* 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.
* 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.