algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

, the nilradical of a

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

is the

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...

consisting of the

nilpotent element In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...

s: :

\mathfrak_R = \lbrace f \in R \mid f^m=0 \text m\in\mathbb_\rbrace.

In the

non-commutative ring In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...

case the same definition does not always work. This has resulted in several

radicals Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...

generalizing the commutative case in distinct ways; see the article

Radical of a ring In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring. The first example of a radical was the nilradical introduced by , based on a suggestion of . In the next few years several other radical ...

for more on this. The

nilradical of a Lie algebra In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical \mathfrak(\mathfrak g) of a finite-dimensional Lie algebra \mathfrak is its maximal nilpotent ideal, which exists because the sum of a ...

is similarly defined for

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

Commutative rings

The nilradical of a commutative ring is the set of all

s in the ring, or equivalently the

radical Radical may refer to: Politics and ideology Politics * Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe an ...

of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent (by the

binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...

), and the product of any element with a nilpotent element is nilpotent (by commutativity). It can also be characterized as the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...

of all the

prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...

s of the ring (in fact, it is the intersection of all

minimal prime ideal In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. Definitio ...

s). A ring is called reduced if it has no nonzero nilpotent. Thus, a ring is reduced

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...

its nilradical is zero. If ''R'' is an arbitrary commutative ring, then the

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of it by the nilradical is a reduced ring and is denoted by

R_

. Since every

maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...

is a prime ideal, the

Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...

— which is the intersection of maximal ideals — must contain the nilradical. A ring ''R'' is called a

Jacobson ring In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which ev ...

if the nilradical and Jacobson radical of ''R''/''P'' coincide for all prime ideals ''P'' of ''R''. An

Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...

is Jacobson, and its nilradical is the maximal

nilpotent ideal In mathematics, more specifically ring theory, an ideal ''I'' of a ring ''R'' is said to be a nilpotent ideal if there exists a natural number ''k'' such that ''I'k'' = 0. By ''I'k'', it is meant the additive subgroup generated by the set of ...

of the ring. In general, if the nilradical is finitely generated (e.g., the ring is

Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...

), then it is

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...

Noncommutative rings

For noncommutative rings, there are several analogues of the nilradical. The lower nilradical (or

Baer Baer (or Bär, from german: bear, links=no) or Van Baer is a surname. Notable people with the surname include: Baer * Alan Baer, American tuba player * Arthur "Bugs" Baer (1886–1969), American journalist and humorist * Buddy Baer (1915–198 ...

–McCoy radical, or prime radical) is the analogue of the radical of the zero ideal and is defined as the intersection of the prime ideals of the ring. The analogue of the set of all nilpotent elements is the upper nilradical and is defined as the ideal generated by all nil ideals of the ring, which is itself a nil ideal. The set of all nilpotent elements itself need not be an ideal (or even a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

), so the upper nilradical can be much smaller than this set. The Levitzki radical is in between and is defined as the largest locally nilpotent ideal. As in the commutative case, when the ring is Artinian, the Levitzki radical is nilpotent and so is the unique largest nilpotent ideal. Indeed, if the ring is merely Noetherian, then the lower, upper, and Levitzki radical are nilpotent and coincide, allowing the nilradical of any Noetherian ring to be defined as the unique largest (left, right, or two-sided) nilpotent ideal of the ring.

References

* Eisenbud, David, "Commutative Algebra with a View Toward Algebraic Geometry", Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, . * Commutative algebra Ideals (ring theory)

Notes

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