# 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, *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 *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 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.

## Notes

- Proof: let be all the (possibly zero) minimal prime ideals.
- Let
*x*be in*D*. Then*xy*= 0 for some nonzero*y*. Since*R*is reduced, (0) is the intersection of all and thus*y*is not in some . Since*xy*is in all ; in particular, in ,*x*is in . - (stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript
*i*. Let .*S*is multiplicatively closed and so we can consider the localization . Let be the pre-image of a maximal ideal. Then is contained in both*D*and and by minimality . (This direction is immediate if*R*is Noetherian by the theory of associated primes.)

- Let
- 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.