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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 Irreducible element ： ウィキペディア英語版 Irreducible element In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units. ==Relationship with prime elements== Irreducible elements should not be confused with prime elements. (A non-zero non-unit element $a$ in a commutative ring $R$ is called prime if, whenever $a\; |\; bc$ for some $b$ and $c$ in $R,$ then $a|b$ or $a|c.)$ In an integral domain, every prime element is irreducible,〔Consider $p$ a prime that is reducible: $p=ab.$ Then $p\; |\; ab\; \backslash Rightarrow\; p\; |\; a$ or $p\; |\; b.$ Say $p\; |\; a\; \backslash Rightarrow\; a\; =\; pc,$ then we have $p=ab=pcb\; \backslash Rightarrow\; p(1-cb)=0.$ Because $R$ is an integral domain we have $cb=1.$ So $b$ is a unit and $p$ is irreducible.〕〔Sharpe (1987) p.54〕 but the converse is not true in general. The converse is true for unique factorization domains〔 (or, more generally, GCD domains.) Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if $D$ is a GCD domain, and $x$ is an irreducible element of $D$, then the ideal generated by $x$ ''is'' a prime ideal of $D$.〔http://planetmath.org/encyclopedia/IrreducibleIdeal.html〕 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「Irreducible element」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.