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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 be all the (possibly zero) minimal prime ideals. : Let ''x'' be in ''D''. Then ''xy'' = 0 for some nonzero ''y''. Since ''R'' is reduced, (0) is the intersection of all and thus ''y'' is not in some . Since ''xy'' is in all ; in particular, in , ''x'' is in . : (stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript ''i''. Let . ''S'' is multiplicatively closed and so we can consider the localization . Let be the pre-image of a maximal ideal. Then is contained in both ''D'' and and by minimality . (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 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
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