Pure submodule
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Let R be a ring, and let M, P be modules over R. If i: P → M is injective then P is a pure submodule of M if, for any R-module X, the natural induced map on tensor products i⊗idX:P⊗X → M⊗X is injective.
Analogously, a short exact sequence
of R-modules is pure exact if the sequence stays exact when tensored with any R-module X. This is equivalent to saying that f(A) is a pure submodule of B.
Purity 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',..., xn' in P such that
Every subspace of a vector space over a field is pure. Every direct summand of M is pure in M. A ring is von Neumann regular if and only if every submodule of every R-module is pure.
If
is a short exact sequence with B being a flat module, 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.
