Dual number

This article deals with a specific type of algebraic structure. For the dual or binary representation of numbers see the article dual system

In the mathematical branch of algebraic geometry , the ring of dual numbers over a body is an algebraic object that is closely related to the concept of the tangential vector .

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .

definition

The dual numbers form a two-dimensional hyper- complex algebra over the field of real numbers. Like the complex numbers , this algebra is generated by two basic elements, the 1 and a non-real unit, which is referred to here with to distinguish it from the imaginary unit of the complex numbers . Every dual number can therefore be clearly identified as ${\displaystyle \mathbb {R} }$ ${\displaystyle i}$ ${\displaystyle \varepsilon }$

${\displaystyle z=a+b\varepsilon }$

with a, b ∈ , i.e. as a linear combination of 1 and . The definition of a general multiplication for dual numbers is completed by a definition for the square of the non-real unit, namely by ${\displaystyle \mathbb {R} }$ ${\displaystyle \varepsilon }$

${\displaystyle \varepsilon ^{2}=0}$.

In addition, as with the complex numbers, the number conjugated to z is

${\displaystyle {\bar {z}}=a-b\varepsilon }$

Are defined.

properties

Like all hyper-complex algebras , the dual numbers also satisfy the right and left-sided distributive law . Like the complex numbers , they are also commutative and associative, and inevitably because there is only one basic element different from 1, namely . ${\displaystyle \varepsilon }$

${\displaystyle 1\cdot \varepsilon =\varepsilon \cdot 1=\varepsilon }$

${\displaystyle 1\cdot (1\cdot \varepsilon )=1\cdot \varepsilon =(1\cdot 1)\cdot \varepsilon =\varepsilon }$

${\displaystyle 1\cdot (\varepsilon \cdot \varepsilon )=1\cdot 0=0=\varepsilon \cdot \varepsilon =(1\cdot \varepsilon )\cdot \varepsilon }$

The dual numbers thus form a commutative ring with one element, which - in contrast to - is not a body, but a main ideal ring with an ideal , namely the real multiple of . The main ideal is that it can be created from a single element . Because of that they are of course zero divisors . ${\displaystyle \mathbb {C} }$ ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon ^{2}=0}$

Matrix display

Since the multiplication of the dual numbers is associative, it can be represented with matrices as follows:

${\displaystyle a+b\varepsilon ~{\mathrel {\widehat {=}}}~{\begin{pmatrix}a&b\\0&a\end{pmatrix}}}$,

what for and especially the nilpotent matrix ${\displaystyle a=0}$ ${\displaystyle b=1}$

${\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}}$

results.

Dual numbers and Laguerre planes

The classic real Laguerre plane can be described (analogous to the description of the classic real Möbius plane using complex numbers ) with the help of the dual numbers (W. Benz: Lectures on the geometry of algebras ).

Algebraic properties

In the terminology of abstract algebra , the dual numbers can be described as the quotient of the polynomial ring and the ideal that is generated by the polynomial , i.e. ${\displaystyle \mathbb {R} [X]}$ ${\displaystyle X^{2}}$

${\displaystyle \mathbb {R} [X]/X^{2}}$.

Dual numbers over rings

Let it be a ring. Then the ring of dual numbers is above the factor ring ${\displaystyle A}$ ${\displaystyle A}$

${\displaystyle A[\varepsilon ]=A[X]/(X^{2});}$

${\displaystyle \varepsilon }$is the image of the indefinite in the quotient ${\displaystyle X}$ ${\displaystyle A[X]/(X^{2}).}$

properties

It is a body . is a local Artinian ring with a vector space over dimension 2. Each element has a unique representation ${\displaystyle k}$ ${\displaystyle k[\varepsilon ]}$ ${\displaystyle k}$

${\displaystyle a+b\varepsilon }$ With ${\displaystyle a,b\in k.}$

The maximum ideal is generated by; is the remainder class field . and are isomorphic as modules. ${\displaystyle \varepsilon }$ ${\displaystyle k}$ ${\displaystyle (\varepsilon )}$ ${\displaystyle k}$ ${\displaystyle k[\varepsilon ]}$

For each ring is ${\displaystyle A}$ ${\displaystyle A[\varepsilon ]\cong A\otimes \mathbb {Z} [\varepsilon ].}$

Dual numbers and derivatives

Suppose a ring, two - algebras and a homomorphism of -algebras. Then there is a natural bijection between ${\displaystyle A}$ ${\displaystyle B,C}$ ${\displaystyle A}$ ${\displaystyle f\colon B\to C}$ ${\displaystyle A}$

the -algebra homomorphisms ${\displaystyle A}$

${\displaystyle B\to C[\varepsilon ],\quad b\mapsto f(b)+\varepsilon D(b),}$

the elevations from below are ${\displaystyle f}$ ${\displaystyle C[\varepsilon ]\to C,}$ ${\displaystyle \varepsilon \mapsto 0,}$

and

${\displaystyle A}$-linear derivations there is the -Modulstruktur on from induced. ${\displaystyle D\colon B\to C;}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle f}$

Significance for algebraic geometry

For a scheme let ${\displaystyle U}$

${\displaystyle U[\varepsilon ]=U\otimes _{\mathbb {Z} }\mathbb {Z} [\varepsilon ].}$

Let it be a scheme and a scheme. The scheme is the relative tangential bundle of over . Then there is natural bijection ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle S}$ ${\displaystyle T_{X/S}=\mathbb {V} (\Omega _{X/S}^{1})=\mathbf {Spec} \,S^{\cdot }\Omega _{X/S}^{1}}$ ${\displaystyle X}$ ${\displaystyle S}$

${\displaystyle T_{X/S}(U)=X(U[\varepsilon ])}$

for any schemes . A - valued point is therefore a - valued point together with a tangential vector at this point. For a body , you can think of it as a point together with a tangential vector. ${\displaystyle S}$ ${\displaystyle U}$ ${\displaystyle U[\varepsilon ]}$ ${\displaystyle U}$ ${\displaystyle \mathrm {Spec} \,k[\varepsilon ]}$ ${\displaystyle k}$

See also

• Abnormal complex number

literature

• Walter Benz : Lectures on the geometry of algebras: geometries by Möbius, Laguerre-Lie, Minkowski in uniform and basic geometric treatment. Springer, 1973, ISBN 978-3-642-88670-6 , p.  21
• M. Demazure, A. Grothendieck: Séminaire de Géométrie algébrique du Bois-Marie. Schemas en groupes I, II, III (SGA 3). Lecture Notes in Mathematics 151, 152, 153. Springer-Verlag, Berlin 1970
• IL Kantor, AS Solodownikow: Hypercomplex numbers. BG Teubner, Leipzig 1978