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 <math>R</math> be a ring (associative, with <math>1</math>), let <math>M</math> be a (left) module over <math>R</math>, and let <math>P</math> be a submodule of <math>M</math> with <math>\iota\colon P\hookrightarrow M</math> be the natural injective map. Then <math>P</math> is a pure submodule of <math>M</math> if, for any (right) <math>R</math>-module <math>X</math>, the natural induced map <math>\mathrm{id}_X\otimes_R \iota\colon X\otimes_R P\to X\otimes_R M</math> is injective.
Analogously, a short exact sequence
:<math>0 \longrightarrow A\,\ \stackrel{f}{\longrightarrow}\ B\,\ \stackrel{g}{\longrightarrow}\ C \longrightarrow 0</math>
of (left) <math>R</math>-modules is pure exact if the sequence stays exact when tensored with any (right) <math>R</math>-module <math>X</math>. This is equivalent to saying that <math>f(A)</math> is a pure submodule of <math>B</math>.
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, <math>P</math> is pure in <math>M</math> if and only if the following condition holds: for any <math>m</math>-by-<math>n</math> matrix <math>(a_{ij})</math> with entries in <math>R</math>, and any set <math>y_1,\cdots,y_m</math> of elements of <math>P</math>, if there exist elements <math>x_1,\cdots,x_n</math> in <math>M</math> such that
:<math>\sum_{j=1}^n a_{ij}x_j = y_i \qquad\mbox{ for } i=1,\ldots,m</math>
then there also exist elements <math>x'_1,\cdots,x'_n</math> in <math>P</math> such that
:<math>\sum_{j=1}^n a_{ij}x'_j = y_i \qquad\mbox{ for } i=1,\ldots,m</math>
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
:<math>0 \longrightarrow A_i \longrightarrow B_i \longrightarrow C_i \longrightarrow 0.</math>
Examples
- Every direct summand of M is pure in M. Consequently, every subspace of a vector space over a field is pure.
Properties
Suppose <math>0 \longrightarrow A\,\ \stackrel{f}{\longrightarrow}\ B\,\ \stackrel{g}{\longrightarrow}\ C \longrightarrow 0</math> is a short exact sequence of <math>R</math>-modules, then:
- <math>C</math> is a flat module if and only if the exact sequence is pure exact for every <math>A</math> and <math>B</math>. From this we can deduce that over a von Neumann regular ring, every submodule of every <math>R</math>-module is pure. This is because every module over a von Neumann regular ring is flat. The converse is also true.
- Suppose <math>B</math> is flat. Then the sequence is pure exact if and only if <math>C</math> is flat. From this one can deduce that pure submodules of flat modules are flat.
- Suppose <math>C</math> is flat. Then <math>B</math> is flat if and only if <math>A</math> is flat.
If <math>0 \longrightarrow A\,\ \stackrel{f}{\longrightarrow}\ B\,\ \stackrel{g}{\longrightarrow}\ C \longrightarrow 0</math> is pure-exact, and <math>F</math> is a finitely presented <math>R</math>-module, then every homomorphism from <math>F</math> to <math>C</math> can be lifted to <math>B</math>, i.e. to every <math>u\colon F\to C</math> there exists <math>v\colon F\to B</math> such that <math>gv=u</math>.