Editing Reduced ring

{{Short description|Ring without non-zero nilpotent elements}}
In [[ring theory]], a branch of [[mathematics]], a [[ring (mathematics)|ring]] 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 (algebra)|square]] zero, that is, ''x''<sup>2</sup>&nbsp;=&nbsp;0 implies ''x''&nbsp;=&nbsp;0. A commutative [[algebra over a ring|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 (ring theory)|ideal]] of ''R'', called the [[nilradical of a ring|nilradical]] of ''R''; therefore a commutative ring is reduced [[if and only if]] its nilradical is [[zero ideal|zero]]. Moreover, a commutative ring is reduced if and only if the only element contained in all [[prime ideal]]s is zero.

A [[quotient ring]] ''R''/''I'' is reduced if and only if ''I'' is a [[radical ideal]].

Let <math>\mathcal{N}_R</math> denote nilradical of a commutative ring <math>R</math>. There is a [[functor]] <math>R \mapsto R/\mathcal{N}_R</math> of the [[category of commutative rings]] <math>\text{Crng}</math> into the [[category (mathematics)|category]] of reduced rings <math>\text{Red}</math> and it is [[left adjoint]] to the inclusion functor <math>I</math> of <math>\text{Red}</math> into <math>\text{Crng}</math>. The natural [[bijection]] <math>\text{Hom}_{\text{Red}}(R/\mathcal{N}_R,S)\cong\text{Hom}_{\text{Crng}}(R,I(S))</math> is induced from the [[universal property]] of quotient rings.

Let ''D'' be the set of all [[zero-divisor]]s in a reduced ring ''R''. Then ''D'' is the [[union (set theory)|union]] of all [[minimal prime ideal]]s.<ref>Proof: let <math>\mathfrak{p}_i</math> be all the (possibly zero) minimal prime ideals.
:<math>D \subset \cup \mathfrak{p}_i:</math> Let ''x'' be in ''D''. Then ''xy'' = 0 for some nonzero ''y''. Since ''R'' is reduced, (0) is the intersection of all <math>\mathfrak{p}_i</math> and thus ''y'' is not in some <math>\mathfrak{p}_i</math>. Since ''xy'' is in all <math>\mathfrak{p}_j</math>; in particular, in <math>\mathfrak{p}_i</math>, ''x'' is in <math>\mathfrak{p}_i</math>.
:<math>D \supset \mathfrak{p}_i:</math> (stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript ''i''. Let <math>S = \{ xy | x \in R - D, y \in R - \mathfrak{p} \}</math>. ''S'' is multiplicatively closed and so we can consider the localization <math>R \to R[S^{-1}]</math>. Let <math>\mathfrak{q}</math> be the pre-image of a maximal ideal. Then <math>\mathfrak{q}</math> is contained in both ''D'' and <math>\mathfrak{p}</math> and by minimality <math>\mathfrak{q} = \mathfrak{p}</math>. (This direction is immediate if ''R'' is Noetherian by the theory of [[associated prime]]s.)</ref>

Over a [[Noetherian ring]] ''R'', we say a [[finitely generated module|finitely generated]] [[module (mathematics)|module]] ''M'' has locally constant rank if <math>\mathfrak{p} \mapsto \operatorname{dim}_{k(\mathfrak{p})}(M \otimes k(\mathfrak{p}))</math> is a locally constant (or equivalently continuous) function on [[Spectrum of a ring|Spec]]&thinsp;''R''. Then ''R'' is reduced if and only if every finitely generated module of locally constant rank is [[projective module|projective]].<ref>{{harvnb|Eisenbud|1995|loc=Exercise 20.13.}}</ref>

==Examples and non-examples==
* [[Subring]]s, [[product of rings|products]], and [[localization of a ring|localizations]] of reduced rings are again reduced rings.
* The ring of [[Integer#Algebraic properties|integers]] '''Z''' is a reduced ring. Every [[Field (mathematics)|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'''[''x'',&thinsp;''y'']/(''xy'') contains ''x'' + (''xy'') and ''y'' + (''xy'') as zero-divisors, but no non-zero nilpotent elements. As another example, the ring '''Z'''&thinsp;×&thinsp;'''Z''' contains (1,&nbsp;0) and (0,&thinsp;1) as zero-divisors, but contains no non-zero nilpotent elements.
* The ring '''Z'''/6'''Z''' is reduced, however '''Z'''/4'''Z''' is not reduced: the class 2&nbsp;+&nbsp;4'''Z''' is nilpotent.  In general, '''Z'''/''n'''''Z''' is reduced if and only if ''n''&nbsp;=&nbsp;0 or ''n'' is [[square-free integer|square-free]].
* If ''R'' is a commutative ring and ''N'' is its [[nilradical of a ring|nilradical]], then the quotient ring ''R''/''N'' is reduced.
* A commutative ring ''R'' of [[prime number|prime]] [[characteristic (algebra)|characteristic]] ''p'' is reduced if and only if its [[Frobenius endomorphism]] is [[injective]] (cf. [[Perfect field]].)

==Generalizations==
Reduced rings play an elementary role in [[algebraic geometry]], where this concept is generalized to the notion of a [[reduced scheme]].

==See also==
*{{section link|Total quotient ring|The total ring of fractions of a reduced ring}}

== Notes ==
{{reflist}}

==References==
* [[Nicolas Bourbaki|N. Bourbaki]], ''Commutative Algebra'', Hermann Paris 1972, Chap. II, § 2.7
* [[Nicolas Bourbaki|N. Bourbaki]], ''Algebra'', Springer 1990, Chap. V, § 6.7
* {{cite book |author-link=David Eisenbud |last=Eisenbud |first=David |title=Commutative Algebra with a View Toward Algebraic Geometry |series=Graduate Texts in Mathematics <!--150?--> |publisher=Springer-Verlag |date=1995 |isbn=0-387-94268-8}}

[[Category:Ring theory]]

[[pl:Element nilpotentny#Pierścień zredukowany]]