Editing Pure submodule

{{Short description|Module components with flexibility in module theory}}
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 (mathematics)|module]].  Pure modules are complementary to [[flat module]]s and generalize Prüfer's notion of [[pure subgroup]]s.  While flat modules are those modules which leave [[short exact sequence]]s exact after [[tensor product|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 module]]s, and a pure exact sequence is a direct limit of [[split exact sequence]]s.

==Definition==

Let ''R'' be a [[ring (mathematics)|ring]] (associative, with 1), let ''M'' be a (left) [[module (mathematics)|module]] over ''R'', let ''P'' be a [[submodule]] of ''M'' and let ''i'': ''P'' →  ''M'' be the natural [[injective]] map. Then ''P'' is a '''pure submodule of ''M''''' if, for any (right) ''R''-module ''X'', the natural induced map id<sub>''X''</sub> ⊗ ''i'' : ''X'' ⊗ ''P'' → ''X'' ⊗ ''M''  (where the [[tensor product]]s are taken over ''R'') is injective.

Analogously, a [[short exact sequence]]
:<math>0 \longrightarrow A\,\ \stackrel{f}{\longrightarrow}\ B\,\ \stackrel{g}{\longrightarrow}\ C \longrightarrow 0</math>
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==
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 (mathematics)|matrix]] (''a''<sub>''ij''</sub>) with entries in ''R'', and any set ''y''<sub>1</sub>, ..., ''y''<sub>''m''</sub> of elements of ''P'', if there exist elements ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> '''in ''M''''' such that
:<math>\sum_{j=1}^n a_{ij}x_j = y_i \qquad\mbox{ for } i=1,\ldots,m</math>
then there also exist elements ''x''<sub>1</sub>′, ..., ''x''<sub>''n''</sub>′ '''in ''P''''' such that
:<math>\sum_{j=1}^n a_{ij}x'_j = y_i \qquad\mbox{ for } i=1,\ldots,m</math>


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 sequence]]s

:<math>0 \longrightarrow  A_i \longrightarrow  B_i \longrightarrow  C_i \longrightarrow  0.</math><ref>For abelian groups, this is proved in {{harvtxt|Fuchs|2015|loc=Ch. 5, Thm. 3.4}}</ref>

==Examples==

* Every [[direct summand]] of ''M'' is pure in ''M''. Consequently, every [[Linear subspace|subspace]] of a [[vector space]] over a [[field (mathematics)|field]] is pure.

==Properties==
Suppose{{sfn|Lam|1999|p=154}}
:<math>0 \longrightarrow A\,\ \stackrel{f}{\longrightarrow}\ B\,\ \stackrel{g}{\longrightarrow}\ C \longrightarrow 0</math>
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.{{sfn|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.


If  <math>0 \longrightarrow A\,\ \stackrel{f}{\longrightarrow}\ B\,\ \stackrel{g}{\longrightarrow}\ C \longrightarrow 0</math>  is pure-exact, and ''F'' is a [[finitely presented module|finitely presented]] ''R''-module, then every homomorphism from ''F'' to ''C'' can be lifted to ''B'', i.e. to every ''u'' : ''F'' → ''C'' there exists ''v'' : ''F'' → ''B'' such that ''gv''=''u''.

==References==
<references/>

*{{Citation|title=Abelian Groups|author=Fuchs|first=László|isbn=9783319194226|series=Springer Monographs in Mathematics|year=2015|publisher=Springer}}

*{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 |mr=1653294 | year=1999}}
[[Category:Module theory]]