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Reduced ring

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.^[1] This property ensures that the only element x satisfying x^n = 0 for some positive integer n is x = 0.^[2] Every commutative ring R admits a canonical reduced quotient R / \mathrm{Nil}(R), where \mathrm{Nil}(R) denotes the ideal of all nilpotent elements, and this quotient inherits many structural properties from R.^[3] A fundamental structural theorem states that any reduced ring embeds as a subdirect product of integral domains, specifically the quotients R / \mathfrak{p} where \mathfrak{p} ranges over the minimal prime ideals of R.^[4] Examples of reduced rings include all integral domains (such as the integers \mathbb{Z}, fields like \mathbb{Q} or \mathbb{C}, and polynomial rings k[x_1, \dots, x_n] over a field k), as well as finite direct products of integral domains; notably, the zero ring is reduced but not an integral domain.^[5] For Noetherian rings, the reduced property is equivalent to satisfying Serre's conditions (R_0) and (S_1), which relate to the depth and dimension of localizations at prime ideals.^[6] Reduced rings are central in algebraic geometry, where an affine scheme \mathrm{Spec}(R) is called reduced if and only if R is reduced, corresponding to schemes without nilpotent elements in their structure sheaf and thus capturing "purely geometric" varieties without infinitesimal structure.^[7] 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 theory.^[8]

Definition

Basic definition

In commutative algebra, a nilpotent element in a ring is a nonzero element a such that a^n = 0 for some integer n > 1.^[9] A commutative ring R with unity is called reduced if it contains no nonzero nilpotent elements; that is, whenever a \in R satisfies a^n = 0 for some integer n > 1, it follows that a = 0.^[9]^[10]^[11] In commutative rings, this condition is equivalent to the statement that x^2 = 0 implies x = 0 for all x \in R. To see this, note that if there exists a nilpotent element of index greater than 2, say a^k = 0 with k > 2 minimal, then b = a^{k-1} satisfies b \neq 0 but b^2 = a^{2k-2} = a^{k} \cdot a^{k-2} = 0, yielding a nonzero square-zero element; conversely, any square-zero element is nilpotent of index 2.^[9] 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.^[10]

Equivalent conditions

A commutative ring R is reduced if and only if its nilradical \mathcal{N}(R) is the zero ideal, where the nilradical \mathcal{N}(R) is defined as the set of all nilpotent elements in R, that is, \mathcal{N}(R) = \{ x \in R \mid x^n = 0 \text{ for some integer } n > 1 \}.^[12]^[13] The nilradical coincides with the intersection of all prime ideals of R, and it is itself a radical ideal, meaning that if y \in R satisfies y^k \in \mathcal{N}(R) for some k \geq 1, then y \in \mathcal{N}(R).^[12]^[13] An equivalent ideal-theoretic characterization is that R is reduced if and only if the zero ideal (0) is a radical ideal: whenever x^n \in (0) for some integer n \geq 1, it follows that x \in (0).^[12] 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 nilpotent precisely when this intersection is trivial.^[13]

Properties

Algebraic properties

In commutative algebra, for any commutative ring R, the quotient R / \mathcal{N}(R) by the nilradical \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 R to a reduced ring factors uniquely through this quotient.^[14] A ring R is reduced if and only if it is isomorphic to its maximal reduced quotient, which occurs precisely when \mathcal{N}(R) = (0).^[15] An ideal I of R is radical if and only if the quotient ring R / I is reduced, since the nilradical of R / I coincides with \sqrt{I} / I, which vanishes exactly when \sqrt{I} = I.^[13] (p. 22) Conversely, if R / I is reduced, then I contains the nilradical \mathcal{N}(R), because the image of \mathcal{N}(R) in R / I would consist of nilpotent elements, which must be zero in a reduced quotient.^[16] The construction R \mapsto R / \mathcal{N}(R) defines a functor from the category of commutative rings to itself that is left adjoint to the inclusion functor of the full subcategory of reduced commutative rings; this reflects the fact that the subcategory of reduced rings is reflective, with the unit of the adjunction being the natural quotient map to the reduced hull. In a reduced ring R, the set of zero-divisors is precisely the union of its minimal prime ideals, as every zero-divisor lies in some minimal prime (since the intersection of all minimal primes is zero) and every element of a minimal prime is a zero-divisor (as the localization at the complement is a field).^[17]

Structural properties

Over a reduced ring R, for a finitely generated projective R-module M, the rank function \rho_M: \operatorname{Spec}(R) \to \mathbb{N}, defined by \mathfrak{p} \mapsto \dim_{\kappa(\mathfrak{p})} (M \otimes_R \kappa(\mathfrak{p})) where \kappa(\mathfrak{p}) is the residue field at \mathfrak{p}, is locally constant with respect to the Zariski topology. In general, over any commutative ring, a finitely generated module is projective if and only if its localization at every prime ideal is free and the rank function is locally constant.^[18] When R is Noetherian and reduced, the Noetherian topology on \operatorname{Spec}(R) ensures that any locally constant function, including the rank function of a finitely generated projective module, takes constant values on each irreducible component of \operatorname{Spec}(R).^[18] Thus, such modules admit a well-defined rank on each irreducible component, reflecting the decomposition of the spectrum into finitely many irreducible components corresponding to the minimal prime ideals. Reduced rings admit a structural decomposition via their minimal prime ideals: R embeds as a subdirect product into \prod R/\mathfrak{p}_i, where the \mathfrak{p}_i are the minimal primes and each R/\mathfrak{p}_i is an integral domain.^[19] This embedding arises because the intersection of the minimal primes is zero (the nilradical), allowing the Chinese remainder-like map to be injective. Reduced rings need not be von Neumann regular, though the classes overlap; for instance, every Boolean ring—where every element is idempotent—is both reduced and von Neumann regular.^[20] Von Neumann regular rings are always reduced, as nilpotent elements would contradict the regularity condition a = a r a for some r.^[21]

Examples and counterexamples

Examples of reduced rings

All integral domains are reduced rings, since the absence of nonzero zero-divisors implies the absence of nonzero nilpotent elements. Classic examples include the ring of integers \mathbb{Z}, which is an integral domain with no nilpotents, and the polynomial ring k over any field k, where the only nilpotent is zero. Similarly, the coordinate ring of an irreducible affine variety over an algebraically closed field is an integral domain, hence reduced.^[22] Direct products of reduced rings are themselves reduced.^[23] For instance, \mathbb{Z} \times \mathbb{Z} is reduced, as neither component introduces nilpotents, though it contains zero-divisors like (1, 0). The same holds for k \times k over a field k. The quotient ring \mathbb{Z}/n\mathbb{Z} is reduced if and only if n is square-free, meaning n has no squared prime factors. An example is \mathbb{Z}/6\mathbb{Z}, which is reduced because $6 = 2 \cdot 3 is square-free; its nilradical is zero, with no nonzero elements whose powers vanish modulo 6. In polynomial rings, quotients by radical ideals yield reduced rings. For example, over a field k, the ring k[x, y]/(xy) is reduced, as the ideal (xy) is generated by a square-free monomial and thus radical, ensuring no nonzero nilpotents in the quotient.^[24] The images of x and y act as orthogonal idempotents, but no element squares to zero nontrivially. Infinite examples abound among function rings. The ring of entire functions on the complex plane forms an integral domain, as the product of two nonzero entire functions is nonzero, making it reduced.^[25] More generally, polynomial rings in any number of variables over a reduced base ring remain reduced, inheriting the nilradical-zero property from the base.^[19]

Examples of non-reduced rings

A classic finite example of a non-reduced commutative ring is \mathbb{Z}/4\mathbb{Z}, where the element $2 + 4\mathbb{Z} satisfies (2 + 4\mathbb{Z})^2 = 0, making it nilpotent of index 2.^[26] In this ring, the nilradical is the principal ideal generated by $2 + 4\mathbb{Z}, which is nonzero.^[27] Another fundamental example arises from polynomial quotients over a field k. The ring k[\varepsilon] = k/(x^2), known as the ring of dual numbers, contains the nonzero element \varepsilon = x + (x^2) such that \varepsilon^2 = 0, hence nilpotent.^[28] This structure illustrates a simple extension where the nilradical is the principal ideal (\varepsilon). For higher nilpotency indices, consider k/(x^3), where the image \overline{x} = x + (x^3) satisfies \overline{x}^3 = 0 but \overline{x} \neq 0 and \overline{x}^2 \neq 0. Here, \overline{x}^2 is nilpotent of index 2, demonstrating a chain of nilpotent elements within the nilradical (\overline{x}). Extending the finite case, \mathbb{Z}/8\mathbb{Z} is non-reduced with nilradical (2\mathbb{Z})/8\mathbb{Z}, generated by $2 + 8\mathbb{Z}, which has index 3 since (2 + 8\mathbb{Z})^3 = 0.^[29] While full matrix rings over fields exhibit nilpotents in the noncommutative setting (e.g., strictly upper triangular matrices), commutative examples like these highlight nonzero nilradicals explicitly. In general, any commutative ring with a nonzero nilradical provides a non-reduced example, particularly Artinian rings whose maximal ideals are nilpotent, such as local Artinian rings where the unique maximal ideal m satisfies m^k = 0 for some k > 0.^[30]

Generalizations

The concept of a reduced ring generalizes to noncommutative rings, where a ring R is defined to be reduced if it contains no nonzero nilpotent elements, that is, whenever a^n = 0 for some positive integer n and a \in R, it follows that a = 0. This property ensures that the ring lacks elements whose powers vanish, mirroring the commutative case but with implications for ideal structure in noncommutative settings. Examples of noncommutative reduced rings include direct products of division rings, since division rings themselves have no nilpotent elements and the direct product operation preserves this absence of nilpotency. In contrast, full matrix rings over division rings, such as the $2 \times 2 matrices over a field, are typically not reduced, as they contain nilpotent elements like strictly upper triangular matrices with nonzero entries. Subrings of products of division rings can also be reduced, providing a broad class for study in noncommutative algebra. Reduced rings in the noncommutative context are closely related to semiprime rings, which are defined as rings with no nonzero nilpotent two-sided ideals. Every reduced ring is semiprime, because the existence of a nilpotent element would generate a nilpotent principal ideal. However, the converse fails: semiprime rings may contain nilpotent elements without forming nilpotent ideals, as seen in matrix rings over reduced rings, where nilpotents exist but the ring has no nilpotent two-sided ideals. This distinction highlights the stricter condition imposed by the reduced property in noncommutative theory. Prime rings, a subclass of semiprime rings, further connect to reduced rings when they lack zero-divisors, but reduced rings need not be prime. The notion extends to algebras over fields, where a k-algebra A (commutative or noncommutative) is reduced if its nilradical—the ideal generated by all nilpotent elements—is zero. This condition is fundamental in representation theory, where reduced algebras ensure that module representations avoid nilpotent actions that could complicate decomposition into irreducibles or semisimple components. For instance, finite-dimensional reduced algebras over algebraically closed fields often decompose into direct sums of simple algebras without nilpotent factors. In universal algebra, the reduced ring concept generalizes to classes of algebras in various signatures that exclude nilpotent elements, forming quasi-varieties defined by quasi-identities such as x^n = 0 \implies x = 0 for each n. These quasi-varieties capture rings without nilpotents and extend to broader algebraic structures, such as those with additional operations, where the absence of nilpotency ensures certain embedding properties or subdirect product decompositions into domains. Unlike full varieties, which are closed under homomorphic images, these classes emphasize the structural rigidity imposed by the reduced condition.

Connections to algebraic geometry

In algebraic geometry, the notion of a reduced ring extends naturally to schemes via the structure sheaf. A scheme X is defined to be reduced if its structure sheaf \mathcal{O}_X is a reduced ring at every point, meaning that for every point x \in X, the local ring \mathcal{O}_{X,x} contains no nonzero nilpotent elements. Equivalently, X is reduced if and only if, for every affine open subset U = \Spec(R) \subset X, the ring of global sections \Gamma(U, \mathcal{O}_X) = R is reduced, ensuring no nonzero nilpotents in \Gamma(V, \mathcal{O}_X) for any open V \subset U. This condition prevents "infinitesimal thickening" or nilpotent structure in the sheaf, mirroring the absence of nilpotents in the algebraic setting.^[31] For an affine scheme X = \Spec(R), reducedness is precisely equivalent to the ring R being reduced. Thus, the spectrum of a reduced ring yields a reduced affine scheme, whose structure sheaf has no nilpotent elements, avoiding embedded points or non-reduced components that could arise from nilpotents in R. This correspondence underscores how reduced rings provide the algebraic foundation for reduced geometric objects, where the prime ideals of R correspond to the points of the scheme without additional nilpotent structure complicating the topology or sheaf. In the context of varieties over an algebraically closed field k, the coordinate ring of an affine variety—defined as k[V] = k[x_1, \dots, x_n]/I(V) for an affine algebraic set V \subset \mathbb{A}^n_k—is reduced if and only if the corresponding scheme is reduced. By the weak Nullstellensatz, the ideal I(V) is radical, so k[V] is always reduced for the classical algebraic set structure; however, in scheme theory, imposing a non-reduced structure (e.g., via a non-radical ideal) introduces nilpotents, corresponding to multiple components or infinitesimal structure not present in the reduced variety. Thus, reduced coordinate rings characterize reduced affine varieties, ensuring the geometric object has no nilpotent "multiplicities."^[32] Normalization further illustrates the interplay between reduced rings and geometry. For a reduced Noetherian scheme X, the normalization \tilde{X} \to X is a birational morphism where \tilde{X} is normal (hence reduced, as normal rings are reduced), preserving the reduced property while resolving singularities. This process, defined via the integral closure in the total ring of fractions, applies specifically to reduced schemes to avoid complications from nilpotents, yielding a normal integral scheme when X is integral. In geometric terms, normalization "unfolds" the reduced variety into a non-singular model without altering its reduced nature.^[33] The integration of reduced rings into algebraic geometry arose in commutative algebra to formalize geometric reducedness, with seminal developments in the 1960s through Alexander Grothendieck's Éléments de géométrie algébrique (EGA). There, reduced sheaves of rings and schemes were introduced to handle infinitesimal structures systematically, enabling the scheme-theoretic framework that unifies classical varieties with modern geometry.^[31]

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