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NilradicalEdit

Nilradical is a central concept in commutative algebra and algebraic geometry. For a ring R, the nilradical Nil(R) consists of all nilpotent elements—those a in R for which a^n = 0 for some positive integer n. In the setting of a commutative ring with unity, Nil(R) has several equivalent characterizations: it is the radical of the zero ideal, sqrt((0)); it is the intersection of all prime ideals of R, i.e., Nil(R) = ⋂{P ⊆ R | P is a prime ideal}; and the quotient R/Nil(R) is a reduced ring (one with no nonzero nilpotent elements). These viewpoints tie together algebraic structure with geometric intuition, a connection that underpins much of modern algebraic geometry and scheme theory.

Historically and mathematically, the nilradical provides a way to measure the extent to which a ring or a geometric object is “nonreduced.” When Nil(R) is nonzero, the corresponding geometric object has infinitesimal thickening or embedded components; when Nil(R) is zero, the object is reduced and, informally, has no such fuzziness. This perspective is crucial for understanding the passage from a ring to its reduced quotient R/Nil(R), which retains the same underlying set of prime ideals (or geometric points) but loses the nilpotent fuzziness.

Definition

  • In a ring R, the nilradical is the set Nil(R) = {a ∈ R | a^n = 0 for some n ≥ 1}. In a commutative context, Nil(R) is an ideal and equals the radical of the zero ideal, sqrt((0)).

  • A standard equivalent description is Nil(R) = ⋂{P | P is a prime ideal of R}. The quotient R/Nil(R) is then a reduced ring.

  • For a subset I ⊆ R, the nilradical satisfies Nil(R/I) ≅ Nil(R)/I, making it a useful tool for studying quotient structures and their geometric interpretations.

Basic properties

  • Nil(R) is an ideal in any commutative ring with 1, and it is a radical ideal (it equals its own radical).

  • R/Nil(R) is reduced, meaning that if x ∈ R dies in the quotient, then x ∈ Nil(R). Equivalently, R/Nil(R) has no nonzero nilpotent elements.

  • If I is any ideal of R, the nilradical of the quotient satisfies Nil(R/I) = Nil(R)/I. This compatibility is important for studying families of rings and their geometric fibers.

  • The elements of Nil(R) are precisely those elements that vanish at every "point" of Spec(R) (in the appropriate geometric sense), but with the added nuance that they contribute infinitesimal structure rather than ordinary points.

Examples

  • In the ring Z of ordinary integers, there are no nonzero nilpotent elements; thus Nil(Z) = 0. The corresponding reduced structure reflects the classical arithmetic without infinitesimal thickening.

  • Consider the quotient ring k[x]/(x^2) over a field k. Here x^2 = 0, so x is nilpotent and Nil(k[x]/(x^2)) = (x). The quotient by the nilradical is k, a reduced ring, illustrating how nilpotents encode a nonreduced, infinitesimal thickening of a single geometric point.

  • A more elaborate example is R = k[x,y]/(x^2, xy, y^2). In this ring, every element a + b x + c y with a ∈ k is nilpotent if and only if a = 0; hence Nil(R) = (x, y). The reduced quotient R/Nil(R) ≅ k is a field, while R itself exhibits nonreduced, embedded structure along the point corresponding to the unique maximal ideal (x, y).

Nilradical and geometry

  • In algebraic geometry, one often studies schemes via their coordinate rings. The nilradical records infinitesimal information. The reduced scheme associated to a ring R is Spec(R/Nil(R)); it has the same underlying topological space as Spec(R), but its structure sheaf reflects a reduced (no nilpotents) algebra.

  • The notion of a “double point” or other infinitesimal thickenings arises from nonzero Nil(R). For example, the spectrum of k[x]/(x^2) represents a single geometric point with a first-order infinitesimal neighborhood.

  • Nonreduced schemes play a role in deformation theory and intersection theory, where nilpotent elements capture first-order deformations and multiplicities. The nilradical helps distinguish purely geometric content (the set of points) from algebraic multiplicities or infinitesimal structure.

Computation and related concepts

  • To compute Nil(R) in a finitely generated algebra over a field, one often uses the fact that Nil(R) is the radical of the zero ideal, i.e., the set of elements that vanish in every prime quotient. Computational tools in computer algebra systems implement methods to determine the radical and, by extension, the nilradical.

  • Related notions include the radical of an ideal I, Rad(I), which consists of all elements r with some power r^n ∈ I. The nilradical Nil(R) is Rad((0)). Understanding these radicals helps connect algebraic properties with geometric interpretations via Spec and the behavior of prime ideals.

Historical notes

  • The idea of a radical of an ideal and its relationship to nilpotent elements has deep roots in the development of modern algebra, with contributions from early 20th-century algebraists establishing the connections between algebraic operations and geometric interpretations. The framing of reduced versus nonreduced structures became central to the language of algebraic geometry as developed in the mid-20th century.

See also