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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 Reduced ring ： ウィキペディア英語版 Reduced ring In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring ''R'' form an ideal of ''R'', called the nilradical of ''R''; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring ''R/I'' is reduced if and only if ''I'' is a radical ideal. Let ''D'' be the set of all zerodivisors in a reduced ring ''R''. Then ''D'' is the union of all minimal prime ideals.〔Proof: let $\backslash mathfrak\_i$ be all the (possibly zero) minimal prime ideals. :$D\; \backslash subset\; \backslash cup\; \backslash mathfrak\_i:$ Let ''x'' be in ''D''. Then ''xy'' = 0 for some nonzero ''y''. Since ''R'' is reduced, (0) is the intersection of all $\backslash mathfrak\_i$ and thus ''y'' is not in some $\backslash mathfrak\_i$. Since ''xy'' is in all $\backslash mathfrak\_j$; in particular, in $\backslash mathfrak\_i$, ''x'' is in $\backslash mathfrak\_i$. :$D\; \backslash supset\; \backslash mathfrak\_i:$ (stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript ''i''. Let $S\; =\; \backslash$. ''S'' is multiplicatively closed and so we can consider the localization $R\; \backslash to\; R()$. Let $\backslash mathfrak$ be the pre-image of a maximal ideal. Then $\backslash mathfrak$ is contained in both ''D'' and $\backslash mathfrak$ and by minimality $\backslash mathfrak\; =\; \backslash mathfrak$. (This direction is immediate if ''R'' is Noetherian by the theory of associated primes.)〕 Over a Noetherian ring ''R'', we say a finitely generated module ''M'' has locally constant rank if $\backslash mathfrak\; \backslash mapsto\; \backslash operatorname\_(M\; \backslash otimes\; k(\backslash mathfrak))$ is a locally constant (or equivalently continuous) function on Spec ''R''. Then ''R'' is reduced if and only if every finitely generated module of locally constant rank is projective. ==Examples and non-examples== * Subrings, products, and localizations of reduced rings are again reduced rings. * The ring of integers Z is a reduced ring. Every field and every polynomial ring over a field (in arbitrarily many variables) is a reduced ring. * More generally, every integral domain is a reduced ring since a nilpotent element is a fortiori a zero divisor. On the other hand, not every reduced ring is an integral domain. For example, the ring Z(''y'' )/(''xy'') contains ''x + (xy)'' and ''y + (xy)'' as zero divisors, but no non-zero nilpotent elements. As another example, the ring Z×Z contains (1,0) and (0,1) as zero divisors, but contains no non-zero nilpotent elements. * The ring Z/6Z is reduced, however Z/4Z is not reduced: The class 2 + 4Z is nilpotent. In general, Z/''n''Z is reduced if and only if ''n'' = 0 or ''n'' is a square-free integer. * If ''R'' is a commutative ring and ''N'' is the nilradical of ''R'', then the quotient ring ''R''/''N'' is reduced. * A commutative ring ''R'' of characteristic ''p'' for some prime number ''p'' is reduced if and only if its Frobenius endomorphism is injective. (cf. perfect field.) 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「Reduced ring」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.