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 (associative, with 1), and let M and 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

0\longrightarrow A\,\ {\stackrel {f}{\longrightarrow }}\ B\,\ {\stackrel {g}{\longrightarrow }}\ C\longrightarrow 0

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

0\longrightarrow A\,\ {\stackrel {f}{\longrightarrow }}\ B\,\ {\stackrel {g}{\longrightarrow }}\ C\longrightarrow 0

is a short exact sequence of R-modules, then:

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)
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.
Suppose C is flat. Then B is flat if and only if A is flat.

Equivalent characterization

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.

^[1]

References

^ For abelian groups, this is proved in Fuchs (2015, Ch. 5, Thm. 3.4)

Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, ISBN 9783319194226

Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294

[1] For abelian groups, this is proved in Fuchs (2015, Ch. 5, Thm. 3.4)

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