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In abstract algebra, specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. If ''N'' is a dense submodule of ''M'', it may alternatively be said that "''N'' ⊆ ''M'' is a rational extension". Dense submodules are connected with rings of quotients in noncommutative ring theory. Most of the results appearing here were first established in , and . It should be noticed that this terminology is different from the notion of a dense subset in general topology. No topology is needed to define a dense submodule, and a dense submodule may or may not be topologically dense in a module with topology. == Definition == This article modifies exposition appearing in and . Let ''R'' be a ring, and ''M'' be a right ''R'' module with submodule ''N''. For an element ''y'' of ''M'', define : Note that the expression ''y''−1 is only formal since it is not meaningful to speak of the module-element ''y'' being invertible, but the notation helps to suggest that ''y''⋅(''y''−1''N'') ⊆ ''N''. The set ''y'' −1''N'' is always a right ideal of ''R''. A submodule ''N'' of ''M'' is said to be a dense submodule if for all ''x'' and ''y'' in ''M'' with ''x'' ≠ 0, there exists an ''r'' in ''R'' such that ''xr'' ≠ and ''yr'' is in ''N''. In other words, using the introduced notation, the set : In this case, the relationship is denoted by : Another equivalent definition is homological in nature: ''N'' is dense in ''M'' if and only if : where ''E''(''M'') is the injective hull of ''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「dense submodule」の詳細全文を読む スポンサード リンク
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