Editing Pure submodule (section)

==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>