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Reduced ring
Reduced ring
Reduced ring
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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 denote nilradical of a commutative ring . There is a functor of the category of commutative rings into the category of reduced rings and it is left adjoint to the inclusion functor of into . The natural bijection 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 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

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  • 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

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

See also

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Notes

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References

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Reduced ring

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In commutative algebra, a reduced ring is a commutative ring with identity that contains no nonzero nilpotent elements, meaning the nilradical of the ring is zero. This property ensures that the only element xx satisfying xn=0x^n = 0 for some positive integer nn is x=0x = 0. Every commutative ring RR admits a canonical reduced quotient R/Nil(R)R / \mathrm{Nil}(R), where Nil(R)\mathrm{Nil}(R) denotes the ideal of all nilpotent elements, and this quotient inherits many structural properties from RR. A fundamental structural theorem states that any reduced ring embeds as a subdirect product of , specifically the quotients R/pR / \mathfrak{p} where p\mathfrak{p} ranges over the minimal prime ideals of RR. Examples of reduced rings include all (such as the integers Z\mathbb{Z}, fields like Q\mathbb{Q} or C\mathbb{C}, and polynomial rings k[x1,…,xn]k[x_1, \dots, x_n] over a field kk), as well as finite direct products of ; in particular, every finite reduced ring is isomorphic to a direct product of finite fields. Notably, the is reduced but not an . For Noetherian rings, the reduced property is equivalent to satisfying Serre's conditions (R0)(R_0) and (S1)(S_1), which relate to the depth and dimension of localizations at prime ideals. Reduced rings are central in , where an affine scheme Spec(R)\mathrm{Spec}(R) is called reduced if and only if RR is reduced, corresponding to schemes without elements in their structure sheaf and thus capturing "purely geometric" varieties without structure. This concept extends to geometrically reduced algebras over fields, which remain reduced after base change to algebraic closures, playing a key role in studying properties like normality and singularities in scheme .

Definition

Basic definition

In , a element in a ring is a nonzero element aa such that an=0a^n = 0 for some n>1n > 1. A RR with unity is called reduced if it contains no nonzero elements; that is, whenever a∈Ra \in R satisfies an=0a^n = 0 for some integer n>1n > 1, it follows that a=0a = 0. In commutative rings, this condition is equivalent to the statement that x2=0x^2 = 0 implies x=0x = 0 for all x∈Rx \in R. To see this, note that if there exists a element of index greater than 2, say ak=0a^k = 0 with k>2k > 2 minimal, then b=ak−1b = a^{k-1} satisfies b≠0b \neq 0 but b2=a2k−2=ak⋅ak−2=0b^2 = a^{2k-2} = a^{k} \cdot a^{k-2} = 0, yielding a nonzero square-zero element; conversely, any square-zero element is of index 2. Although the notion of reduced rings is primarily developed in the context of commutative rings with unity, the concept extends to noncommutative rings by retaining the condition of having no nonzero nilpotent elements, albeit with modifications to certain equivalent characterizations and properties that rely on commutativity.

Equivalent conditions

A commutative ring RR is reduced if and only if its nilradical N(R)\mathcal{N}(R) is the zero ideal, where the nilradical N(R)\mathcal{N}(R) is defined as the set of all nilpotent elements in RR, that is, N(R)={x∈R∣xn=0 for some integer n>1}\mathcal{N}(R) = \{ x \in R \mid x^n = 0 \text{ for some integer } n > 1 \}. The nilradical coincides with the intersection of all prime ideals of RR, and it is itself a radical ideal, meaning that if y∈Ry \in R satisfies yk∈N(R)y^k \in \mathcal{N}(R) for some k≥1k \geq 1, then y∈N(R)y \in \mathcal{N}(R). An equivalent ideal-theoretic characterization is that RR is reduced if and only if the zero ideal (0)(0) is a radical ideal: whenever xn∈(0)x^n \in (0) for some n≥1n \geq 1, it follows that x∈(0)x \in (0). This condition leverages the role of prime ideals in the nilradical, as the intersection of all such primes precisely captures the nilpotents, ensuring that no nonzero element is precisely when this intersection is trivial.

Properties

Algebraic properties

In commutative algebra, for any commutative ring RR, the quotient R/N(R)R / \mathcal{N}(R) by the nilradical N(R)\mathcal{N}(R) is reduced, as it eliminates all nilpotent elements, and it serves as the maximal reduced quotient in the sense that any surjective ring homomorphism from RR to a reduced ring factors uniquely through this quotient. A ring RR is reduced if and only if it is isomorphic to its maximal reduced quotient, which occurs precisely when N(R)=(0)\mathcal{N}(R) = (0). An ideal II of RR is radical if and only if the quotient ring R/IR / I is reduced, since the nilradical of R/IR / I coincides with I/I\sqrt{I} / I
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