# 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, *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 [math]\displaystyle{ \mathcal{N}_R }[/math] be nilradical of any commutative ring [math]\displaystyle{ R }[/math]. There is a natural functor [math]\displaystyle{ R\mapsto R/\mathcal{N}_R }[/math] of category of commutative rings [math]\displaystyle{ \text{Crng} }[/math] into category of reduced rings [math]\displaystyle{ \text{Red} }[/math] and it is left adjoint to the inclusion functor [math]\displaystyle{ I }[/math] of [math]\displaystyle{ \text{Red} }[/math] into [math]\displaystyle{ \text{Crng} }[/math] . The bijection [math]\displaystyle{ \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-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 [math]\displaystyle{ \mathfrak{p} \mapsto \operatorname{dim}_{k(\mathfrak{p})}(M \otimes k(\mathfrak{p})) }[/math] 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**/6**Z**is reduced, however**Z**/4**Z**is not reduced: The class 2 + 4**Z**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.)

## Generalizations

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

## See also

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

## Notes

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

- [math]\displaystyle{ D \subset \cup \mathfrak{p}_i: }[/math] Let
- ↑ 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

Original source: https://en.wikipedia.org/wiki/Reduced ring.
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