cid fc50a8a9b4e09ab0475ad8901e298fb502fd066a8e1a32ab9cce4b67fafc0ca7 · original · open access · via wikipedia
Title: Pure submodule Authors: Venue: Wikipedia Year: unknown Canonical: wiki:pure-submodule URL: https://en.wikipedia.org/wiki/Pure_submodule Collector: wikipedia OA: true ABSTRACT 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 limit of projective modules, and a pure exact sequence is a direct limit of split exact sequences.