Nilpotent

Last updated January 28, 2024

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 $x^{n}=0$ .

Examples

This definition can be applied in particular to square matrices. The matrix

A={\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}}

is nilpotent because

A^{3}=0

. See nilpotent matrix for more.

In the factor ring $\mathbb {Z} /9\mathbb {Z}$ , the equivalence class of 3 is nilpotent because 3² is congruent to 0 modulo 9.
Assume that two elements $a$ and $b$ in a ring $R$ satisfy $ab=0$ . Then the element $c=ba$ is nilpotent as ${\begin{aligned}c^{2}&=(ba)^{2}\\&=b(ab)a\\&=0.\\\end{aligned}}$ An example with matrices (for a, b): $A={\begin{pmatrix}0&1\\0&1\end{pmatrix}},\;\;B={\begin{pmatrix}0&1\\0&0\end{pmatrix}}.$ Here $AB=0$ and $BA=B$ .

By definition, any element of a nilsemigroup is nilpotent.

Properties

No nilpotent element can be a unit (except in the trivial ring, which has only a single element 0 = 1). All nilpotent elements are zero divisors.

An $n\times n$ matrix $A$ with entries from a field is nilpotent if and only if its characteristic polynomial is $t^{n}$ .

If $x$ is nilpotent, then $1-x$ is a unit, because $x^{n}=0$ entails

(1-x)(1+x+x^{2}+\cdots +x^{n-1})=1-x^{n}=1.

More generally, the sum of a unit element and a nilpotent element is a unit when they commute.

Commutative rings

The nilpotent elements from a commutative ring $R$ form an ideal ${\mathfrak {N}}$ ; this is a consequence of the binomial theorem. This ideal is the nilradical of the ring. Every nilpotent element $x$ in a commutative ring is contained in every prime ideal ${\mathfrak {p}}$ of that ring, since $x^{n}=0\in {\mathfrak {p}}$ . So ${\mathfrak {N}}$ is contained in the intersection of all prime ideals.

If $x$ is not nilpotent, we are able to localize with respect to the powers of $x$ : $S=\{1,x,x^{2},...\}$ to get a non-zero ring $S^{-1}R$ . The prime ideals of the localized ring correspond exactly to those prime ideals ${\mathfrak {p}}$ of $R$ with ${\mathfrak {p}}\cap S=\emptyset$ .^[2] As every non-zero commutative ring has a maximal ideal, which is prime, every non-nilpotent $x$ is not contained in some prime ideal. Thus ${\mathfrak {N}}$ is exactly the intersection of all prime ideals.^[3]

A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of ring $R$ are precisely those that annihilate all integral domains internal to the ring $R$ (that is, of the form $R/I$ for prime ideals $I$ ). This follows from the fact that nilradical is the intersection of all prime ideals.

Nilpotent elements in Lie algebra

Let ${\mathfrak {g}}$ be a Lie algebra. Then an element $x\in {\mathfrak {g}}$ is called nilpotent if it is in $[{\mathfrak {g}},{\mathfrak {g}}]$ and $\operatorname {ad} x$ is a nilpotent transformation. See also: Jordan decomposition in a Lie algebra.

Nilpotency in physics

Any ladder operator in a finite dimensional space is nilpotent. They represent creation and annihilation operators, which transform from one state to another, for example the raising and lowering Pauli matrices $\sigma _{\pm }=(\sigma _{x}\pm i\sigma _{y})/2$ .

An operand $Q$ that satisfies $Q^{2}=0$ is nilpotent. Grassmann numbers which allow a path integral representation for Fermionic fields are nilpotents since their squares vanish. The BRST charge is an important example in physics.

As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition.^[4]^[5] More generally, in view of the above definitions, an operator $Q$ is nilpotent if there is $n\in \mathbb {N}$ such that $Q^{n}=0$ (the zero function). Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with $n=2$ ). Both are linked, also through supersymmetry and Morse theory,^[6] as shown by Edward Witten in a celebrated article.^[7]

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.^[8] More generally, the technique of microadditivity (which can used to derive theorems in physics) makes use of nilpotent or nilsquare infinitesimals and is part smooth infinitesimal analysis.

Algebraic nilpotents

The two-dimensional dual numbers contain a nilpotent space. Other algebras and numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions $\mathbb {C} \otimes \mathbb {H}$ , and complex octonions $\mathbb {C} \otimes \mathbb {O}$ . If a nilpotent infinitesimal is a variable tending to zero, it can be shown that any sum of terms for which it is the subject is an indefinitely small proportion of the first order term.

Related Research Articles

In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

<span class="mw-page-title-main">Lie algebra</span> Algebraic structure used in analysis

In mathematics, a Lie algebra is a vector space $together with an operation called the Lie bracket, an alternating bilinear map, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, with the Lie bracket defined as the commutator .$

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form $a + bε$ , where $a$ and $b$ are real numbers, and $ε$ is a symbol taken to satisfy $with .$

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In abstract algebra, an element $a$ of a ring $R$ is called a left zero divisor if there exists a nonzero $x$ in $R$ such that $ax = 0$ , or equivalently if the map from $R$ to $R$ that sends $x$ to $ax$ is not injective. Similarly, an element $a$ of a ring is called a right zero divisor if there exists a nonzero $y$ in $R$ such that $ya = 0$ . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element $a$ that is both a left and a right zero divisor is called a two-sided zero divisor. If the ring is commutative, then the left and right zero divisors are the same.

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

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 yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by J(R) or rad(R); the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in Jacobson 1945.

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 specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

In ring theory, a branch of mathematics, the radical of an ideal $of a commutative ring is another ideal defined by the property that an element is in the radical if and only if some power of is in . Taking the radical of an ideal is called radicalization . A radical ideal is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal.$

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements:

In mathematics, the Heisenberg group $, named after Werner Heisenberg, is the group of 3\times3 upper triangular matrices of the form$

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.

In abstract algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x⁻¹ belongs to D.

In mathematics, a unipotent elementr of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)ⁿ is zero for some n.

In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yⁿ is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pⁿ) is a primary ideal if p is a prime number.

In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.

In this article, we discuss certain applications of the dual quaternion algebra to 2D geometry. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which we will call the planar quaternions.

References

↑ Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127.
↑ Matsumura, Hideyuki (1970). "Chapter 1: Elementary Results". Commutative Algebra. W. A. Benjamin. p. 6. ISBN 978-0-805-37025-6.
↑ Atiyah, M. F.; MacDonald, I. G. (February 21, 1994). "Chapter 1: Rings and Ideals". Introduction to Commutative Algebra. Westview Press. p. 5. ISBN 978-0-201-40751-8.
↑ Peirce, B. Linear Associative Algebra. 1870.
↑ Polcino Milies, César; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0
↑ A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714, 2000 doi : 10.1088/0264-9381/17/18/309.
↑ E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982.
↑ Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, ISBN 978-981-270-914-1

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Polcino Milies & Sehgal (2002), An Introduction to Group Rings. p. 127.

[2] Matsumura, Hideyuki (1970). "Chapter 1: Elementary Results". Commutative Algebra. W. A. Benjamin. p. 6. ISBN 978-0-805-37025-6.

[3] Atiyah, M. F.; MacDonald, I. G. (February 21, 1994). "Chapter 1: Rings and Ideals". Introduction to Commutative Algebra. Westview Press. p. 5. ISBN 978-0-201-40751-8.

[4] Peirce, B. Linear Associative Algebra. 1870.

[5] Polcino Milies, César; Sehgal, Sudarshan K. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0

[6] A. Rogers, The topological particle and Morse theory, Class. Quantum Grav. 17:3703–3714, 2000 doi : 10.1088/0264-9381/17/18/309.

[7] E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982.

[8] Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, ISBN 978-981-270-914-1

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]