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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pure submodule ： ウィキペディア英語版 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 limit 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 :$\backslash sum\_^n\; a\_x\_j\; =\; y\_i\; \backslash qquad\backslash mbox\; i=1,\backslash ldots,m$ then there also exist elements ''x''1',..., ''x''''n''' in ''P'' such that :$\backslash sum\_^n\; a\_x\text{'}\_j\; =\; y\_i\; \backslash qquad\backslash mbox\; i=1,\backslash ldots,m$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pure submodule」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.