# Reduced ring

In ring theory, a branch of mathematics, a 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 zero, that is, x2 = 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 $\displaystyle{ \mathcal{N}_R }$ be nilradical of any commutative ring $\displaystyle{ R }$. There is a natural functor $\displaystyle{ R\mapsto R/\mathcal{N}_R }$ of category of commutative rings $\displaystyle{ \text{Crng} }$ into category of reduced rings $\displaystyle{ \text{Red} }$ and it is left adjoint to the inclusion functor $\displaystyle{ I }$ of $\displaystyle{ \text{Red} }$ into $\displaystyle{ \text{Crng} }$ . The bijection $\displaystyle{ \text{Hom}_{\text{Red}}(R/\mathcal{N}_R,S)\cong\text{Hom}_{\text{Crng}}(R,I(S)) }$ is induced from the universal property of quotient rings.

Let D be the set of all zero-divisors in a reduced ring R. Then D is the union of all minimal prime ideals.[1]

Over a Noetherian ring R, we say a finitely generated module M has locally constant rank if $\displaystyle{ \mathfrak{p} \mapsto \operatorname{dim}_{k(\mathfrak{p})}(M \otimes k(\mathfrak{p})) }$ 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.[2]

## 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[x, 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/nZ 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.)

## Generalizations

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

• Total quotient ring#The total ring of fractions of a reduced ring

## Notes

1. Proof: let $\displaystyle{ \mathfrak{p}_i }$ be all the (possibly zero) minimal prime ideals.
$\displaystyle{ D \subset \cup \mathfrak{p}_i: }$ Let x be in D. Then xy = 0 for some nonzero y. Since R is reduced, (0) is the intersection of all $\displaystyle{ \mathfrak{p}_i }$ and thus y is not in some $\displaystyle{ \mathfrak{p}_i }$. Since xy is in all $\displaystyle{ \mathfrak{p}_j }$; in particular, in $\displaystyle{ \mathfrak{p}_i }$, x is in $\displaystyle{ \mathfrak{p}_i }$.
$\displaystyle{ D \supset \mathfrak{p}_i: }$ (stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript i. Let $\displaystyle{ S = \{ xy | x \in R - D, y \in R - \mathfrak{p} \} }$. S is multiplicatively closed and so we can consider the localization $\displaystyle{ R \to R[S^{-1}] }$. Let $\displaystyle{ \mathfrak{q} }$ be the pre-image of a maximal ideal. Then $\displaystyle{ \mathfrak{q} }$ is contained in both D and $\displaystyle{ \mathfrak{p} }$ and by minimality $\displaystyle{ \mathfrak{q} = \mathfrak{p} }$. (This direction is immediate if R is Noetherian by the theory of associated primes.)
2. Eisenbud 1995, Exercise 20.13.

## References

• N. Bourbaki, Commutative Algebra, Hermann Paris 1972, Chap. II, § 2.7
• N. Bourbaki, Algebra, Springer 1990, Chap. V, § 6.7
• Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Springer-Verlag. ISBN 0-387-94268-8.

pl:Element nilpotentny#Pierścień zredukowany