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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 essential extension ： ウィキペディア英語版 essential extension In mathematics, specifically module theory, given a ring ''R'' and ''R''-modules ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ''M'') if for every submodule ''H'' of ''M'', :$H\backslash cap\; N=\backslash \backslash ,$ implies that $H=\backslash \backslash ,$ As a special case, an essential left ideal of ''R'' is a left ideal which is essential as a submodule of the left module ''R''''R''. The left ideal has non-zero intersection with any non-zero left ideal of ''R''. Analogously, and essential right ideal is exactly an essential submodule of the right ''R'' module ''R''''R'' The usual notations for essential extensions include the following two expressions: :$N\backslash subseteq\_e\; M\backslash ,$ , and $N\backslash trianglelefteq\; M$ The dual notion of an essential submodule is that of superfluous submodule (or small submodule). A submodule ''N'' is superfluous if for any other submodule ''H'', :$N+H=M\backslash ,$ implies that $H=M\backslash ,$. The usual notations for superfluous submodules include: :$N\backslash subseteq\_s\; M\backslash ,$ , and $N\backslash ll\; M$ ==Properties== Here are some of the elementary properties of essential extensions, given in the notation introduced above. Let ''M'' be a module, and ''K'', ''N'' and ''H'' be submodules of ''M'' with ''K'' $\backslash subset$ ''N'' *Clearly ''M'' is an essential submodule of ''M'', and the zero submodule of a nonzero module is never essential. *$K\backslash subseteq\_e\; M$ if and only if $K\backslash subseteq\_e\; N$ and $N\backslash subseteq\_e\; M$ *$K\; \backslash cap\; H\; \backslash subseteq\_e\; M$ if and only if $K\backslash subseteq\_e\; M$ and $H\backslash subseteq\_e\; M$ Using Zorn's Lemma it is possible to prove another useful fact: For any submodule ''N'' of ''M'', there exists a submodule ''C'' such that :$N\backslash oplus\; C\; \backslash subseteq\_e\; M$. Furthermore, a module with no proper essential extension (that is, if the module is essential in another module, then it is equal to that module) is an injective module. It is then possible to prove that every module ''M'' has a maximal essential extension ''E''(''M''), called the injective hull of ''M''. The injective hull is necessarily an injective module, and is unique up to isomorphism. The injective hull is also minimal in the sense that any other injective module containing ''M'' contains a copy of ''E''(''M''). Many properties dualize to superfluous submodules, but not everything. Again with let ''M'' be a module, and ''K'', ''N'' and ''H'' be submodules of ''M'' with ''K'' subset ''N''. *The zero submodule is always superfluous, and a nonzero module ''M'' is never superfluous in itself. *$N\backslash subseteq\_s\; M$ if and only if $K\backslash subseteq\_s\; M$ and $N/K\; \backslash subseteq\_s\; M/K$ *$K+H\backslash subseteq\_s\; M$ if and only if $K\backslash subseteq\_s\; M$ and $H\backslash subseteq\_s\; M$. Since every module can be mapped via a monomorphism whose image is essential in an injective module (its injective hull), one might ask if the dual statement is true, i.e. for every module ''M'', is there a projective module ''P'' and an epimorphism from ''P'' onto ''M'' whose kernel is superfluous? (Such a ''P'' is called a projective cover). The answer is "''No''" in general, and the special class of rings which provide their right modules projective covers is the class of right perfect rings. 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「essential extension」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.