# 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

[edit]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

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

[edit]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

then there also exist elements *x*_{1}′, ..., *x*_{n}′ **in P** 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

^{[1]}

## Examples

[edit]- Every direct summand of
*M*is pure in*M*. Consequently, every subspace of a vector space over a field is pure.

## Properties

[edit]Suppose^{[2]}

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.^{[3]}- 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 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

[edit]- Fuchs, László (2015),
*Abelian Groups*, Springer Monographs in Mathematics, Springer, ISBN 9783319194226

- Lam, Tsit-Yuen (1999),
*Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294