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

be a ring (associative, with

), let

be a (left) module over

, and let

be a submodule of

with

\iota\colonP\hookrightarrowM

be the natural injective map. Then

is a pure submodule of
M

if, for any (right)

-module

, the natural induced map

id_{X ⊗}_R\iota\colonX ⊗ _RP\toX ⊗ _RM

is injective.

Analogously, a short exact sequence

0\longrightarrowA \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C\longrightarrow0

of (left)

-modules is pure exact if the sequence stays exact when tensored with any (right)

-module

. This is equivalent to saying that

f(A)

is a pure submodule of

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,

is pure in

if and only if the following condition holds: for any

-by-

matrix

(a_ij)

with entries in

, and any set

y_{1, … ,y}_m

of elements of

, if there exist elements

x_{1, … ,x}_n

in
M

such that

	n
\sum
	j=1

a_ijx_j=y_i fori=1,\ldots,m

then there also exist elements

x'_{1, … ,x'}_n

in
P

such that

	n
\sum
	j=1

a_ijx'_j=y_i fori=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\longrightarrowA_i\longrightarrowB_i\longrightarrowC_i\longrightarrow0.

^[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\longrightarrowA \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C\longrightarrow0

is a short exact sequence of

-modules, then:

is a flat module if and only if the exact sequence is pure exact for every

and

. 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.

Suppose

is flat. Then the sequence is pure exact if and only if

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

Suppose

is flat. Then

is flat if and only if

is flat.

0\longrightarrowA \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C\longrightarrow0

is pure-exact, and

is a finitely presented

-module, then every homomorphism from

can be lifted to

, i.e. to every

u\colonF\toC

there exists

v\colonF\toB

such that

gv=u

References

For abelian groups, this is proved in