Pure submodule

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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.

Definition

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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: PM be the natural injective map. Then P is a pure submodule of M if, for any (right) R-module X, the natural induced map idXi : XPXM (where the tensor products are taken over R) is injective.

Analogously, a short exact sequence

 

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

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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 (aij) with entries in R, and any set y1, ..., ym of elements of P, if there exist elements x1, ..., xn in M such that

 

then there also exist elements x1′, ..., xnin P 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]

Examples

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Properties

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Suppose[2]

 

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.[3]
  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   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 : FC there exists v : FB such that gv=u.

References

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  1. ^ For abelian groups, this is proved in Fuchs (2015, Ch. 5, Thm. 3.4)
  2. ^ Lam 1999, p. 154.
  3. ^ Lam 1999, p. 162.
  • Fuchs, László (2015), Abelian Groups, Springer Monographs in Mathematics, Springer, ISBN 9783319194226