Pure submodule

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In abstract algebra, a submodule P of a module M over some ring R is pure in M<b> if for any R-module X, the naturally induced map of tensor products P⊗X → M⊗X is injective.

Analogously, a short exact sequence

Image:Short exact sequence ABC.png

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 (a_ij) with entries in R, and any set y₁,...,y_m of elements of P, if there exist elements x₁,...,x_n <b>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 x₁',..., x_n' in P such that

<math>\sum_{j=1}^n a_{ij}x'_j = y_i \qquad\mbox{ for } i=1,\ldots,m</math>

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

Image:Short exact sequence ABC.png

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.