# Reduced ring

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, 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 D be the set of all zerodivisors 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.

## 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, 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, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.