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.

Properties

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 $𝔑_{R}$ denote the nilradical of a commutative ring $R$ . There is a functor $R \mapsto R / 𝔑_{R}$ of the category of commutative rings $𝖢𝖱𝗂𝗇𝗀$ into the category of reduced rings $𝖱𝖾𝖽$ and it is left adjoint to the inclusion functor $I$ of $𝖱𝖾𝖽$ into $𝖢𝖱𝗂𝗇𝗀$ , making $𝖱𝖾𝖽$ a reflective subcategory of $𝖢𝖱𝗂𝗇𝗀$ . The natural bijection ${Hom}_{𝖱𝖾𝖽} (R / 𝔑_{R}, S) ≅ {Hom}_{𝖢𝖱𝗂𝗇𝗀} (R, I (S))$ 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.

Over a Noetherian ring R, we say a finitely generated module M has locally constant rank if $𝔭 \mapsto \dim_{k (𝔭)} (M \otimes k (𝔭))$ 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.

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 square-free.
If R is a commutative ring and N is its nilradical, then the quotient ring R/N is reduced.
A commutative ring R of prime characteristic 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 notion of a reduced scheme.

Notes

↑ Proof: let $𝔭_{i}$ be all the (possibly zero) minimal prime ideals.
$D \subset \cup 𝔭_{i} :$ Let x be in D. Then xy = 0 for some nonzero y. Since R is reduced, (0) is the intersection of all $𝔭_{i}$ and thus y is not in some $𝔭_{i}$ . Since xy is in all $𝔭_{j}$ ; in particular, in $𝔭_{i}$ , x is in $𝔭_{i}$ .

$D \supset 𝔭_{i} :$ (stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript i. Let $S = {x y | x \in R - D, y \in R - 𝔭}$ . S is multiplicatively closed and so we can consider the localization $R \to R [S^{- 1}]$ . Let $𝔮$ be the pre-image of a maximal ideal. Then $𝔮$ is contained in both D and $𝔭$ and by minimality $𝔮 = 𝔭$ . (This direction is immediate if R is Noetherian by the theory of associated primes.)
↑ 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.

[1] Proof: let $𝔭_{i}$ be all the (possibly zero) minimal prime ideals.
$D \subset \cup 𝔭_{i} :$ Let x be in D. Then xy = 0 for some nonzero y. Since R is reduced, (0) is the intersection of all $𝔭_{i}$ and thus y is not in some $𝔭_{i}$ . Since xy is in all $𝔭_{j}$ ; in particular, in $𝔭_{i}$ , x is in $𝔭_{i}$ .

$D \supset 𝔭_{i} :$ (stolen from Kaplansky, commutative rings, Theorem 84). We drop the subscript i. Let $S = {x y | x \in R - D, y \in R - 𝔭}$ . S is multiplicatively closed and so we can consider the localization $R \to R [S^{- 1}]$ . Let $𝔮$ be the pre-image of a maximal ideal. Then $𝔮$ is contained in both D and $𝔭$ and by minimality $𝔮 = 𝔭$ . (This direction is immediate if R is Noetherian by the theory of associated primes.)

[2] Eisenbud 1995, Exercise 20.13.