Pure submodule

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In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave short exact sequences exact after tensoring, a pure submodule defines a short exact sequence that remains exact after tensoring with any module. Similarly a flat module is a direct limits of projective modules, and a pure submodule defines a short exact sequence which is a direct limit of split exact sequences, each defined by a direct summand.

[edit] Definition

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

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 (aij) with entries in R, and any set y1,...,ym of elements of P, if there exist elements x1,...,xn 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',..., xn' in P such that

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

[edit] Examples

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.

[edit] References