Pure submodule

In mathematics, pure submodules are a concept complementary to flat modules and generalize Pruefer'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 modules are precisely the direct limits of projective modules, and a pure submodule defines a short exact sequence which is a direct limit of split exact sequences.

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 subspace of a vector space over a field is pure. Every direct summand of M is pure in M. A ring is von Neumann regular if and only if every submodule of every R-module is pure.

If

File:Short exact sequence ABC.png

is a short exact sequence with B being a flat module, 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.

References

• Lam, Tsit-Yuen (1999). Lectures on Modules and Rings. Springer. ISBN 0-387-98428-3.