Editing Pure submodule (section)

==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''.