Module theory

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 (known as a pure exact sequence) that remains exact after tensoring with any module. Similarly a flat module is a direct limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences. (Wikipedia).

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Related pages

Projective module | Split exact sequence | Flat module | Linear subspace | Vector space | Mathematics | Pure subgroup | Von Neumann regular ring | Field (mathematics) | Tensor product | Matrix (mathematics) | Ring (mathematics) | Direct limit | Module (mathematics)