Pure submodule

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

Analogously, a short exact sequence

File: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 elments x₁,...,x_n in M such that

$\sum _{j=1}^{n}a_{ij}x_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m$

then there also exist elements x₁,..., x_n in P such that

$\sum _{j=1}^{n}a_{ij}x_{j}=y_{i}\qquad {\mbox{ for }}i=1,\ldots ,m$

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.