Pure submodule

Pure submodule

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.

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"⊗id"X":"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 ("a""ij") with entries in "R", and any set "y"1,...,"y""m" of elements of "P", if there exist elements "x"1,...,"x""n" in "M" such that:sum_{j=1}^n a_{ij}x_j = y_i qquadmbox{ for } i=1,ldots,mthen there also exist elements "x"1',..., "x""n"' in "P" such that:sum_{j=1}^n a_{ij}x'_j = y_i qquadmbox{ for } i=1,ldots,m

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: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.

References

*cite book | first = Tsit-Yuen | last = Lam | year = 1999 | title = Lectures on Modules and Rings | publisher = Springer | id = ISBN 0-387-98428-3


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