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 that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure submodule defines a short exact sequence which is a direct limit of split exact sequences, each defined by a direct summand.

Definition

Let R be a ring, and let M, P be modules over R. If i: P → M is injective then P is a pure submodule of M if, for any R-module X, the natural induced map on tensor products i⊗id_X:P⊗X → M⊗X is injective.

Analogously, a short exact sequence

File:Short exact sequence ABC.png

of R-modules is pure exact if the sequence stays exact when tensored with any R-module X. This is equivalent to saying that f(A) is a pure submodule of B.

Purity 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

$\sum_{j=1}^n a_{ij}x'_j = y_i \qquad\mbox{ for } i=1,\ldots,m$

Examples

• Every direct summand of M is pure in M. Consequently, every subspace of a vector space over a field is pure.

• (Lam 1999, p.154) Suppose

File:Short exact sequence ABC.png

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. (Lam 1999, p.162)
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.