Pure submodule

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 idX* ⊗ 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_{j=1}^n a_{ij}x_j = y_i \qquad\mbox{ for } i=1,\ldots,m then there also exist elements x1′, ..., 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:

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

References

1. For abelian groups, this is proved in {{harvtxt. Fuchs. 2015