Editing Reduced ring (section)

{{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>