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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.

Let be a ring (associative, with ), let be a (left) module over , and let be a submodule of with be the natural injective map. Then is a pure submodule of if, for any (right) -module , the natural induced map is injective.

Analogously, a short exact sequence

of (left) -modules is pure exact if the sequence stays exact when tensored with any (right) -module . This is equivalent to saying that is a pure submodule of .

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, is pure in if and only if the following condition holds: for any -by- matrix with entries in , and any set of elements of , if there exist elements in such that

then there also exist elements in such that

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

^[1]

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

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

1. is a flat module if and only if the exact sequence is pure exact for every and . From this we can deduce that over a von Neumann regular ring, every submodule of every -module is pure. This is because every module over a von Neumann regular ring is flat. The converse is also true.^[2]
2. Suppose is flat. Then the sequence is pure exact if and only if is flat. From this one can deduce that pure submodules of flat modules are flat.
3. Suppose is flat. Then is flat if and only if is flat.

If is pure-exact, and is a finitely presented -module, then every homomorphism from to can be lifted to , i.e. to every there exists such that .

• 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