Dual number

In the mathematical subfield of algebraic geometry is the ring of dual numbers over a field is an algebraic object that is closely related to the notion of the tangent vector.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. For more details see Commutative Algebra.

Definition

The dual numbers form a two-dimensional hypercomplex algebra over the field of real numbers; The complex numbers as this algebra is generated by two basic elements 1 and a non- real unit which is referred to herein with numbers to distinguish from the imaginary component of the complex. Each dual number can be so clear as

With a, b ​​∈ represent, ie 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

In addition, as with the complex numbers to z konjuguierte number

Defined.

Properties

Like all hypercomplex algebras also satisfy the dual numbers, the right - and left- distributive. Like the complex numbers, they are also commutative and associative, and indeed necessarily, since there is only one different from the one base element, namely.

Thus, the dual numbers form a commutative ring with identity, but - in contrast to - not a body, but a principal ideal ring with an ideal, namely the reellzahligen multiples of. Ever ideal, since it can be generated by a single element. Because they are naturally zero divisor.

Matrix representation

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

Which for a = 0 and b = 1 is precisely the nilpotent matrix

Results.

Algebraic properties

In the terminology of abstract algebra, the dual numbers as the quotient of the polynomial ring and the ideal can be described, which is generated by the polynomial, ie

Dual numbers rings

It was a ring. Then the ring of dual numbers over the factor ring

Is the image of the indeterminates in the quotient

Properties

It is a body. is a local artinian ring having a vector space of dimension 2. Each element has a unique representation

The maximal ideal is generated by; the residue field is. and are as -modules isomorphic.

For each ring

Dual numbers and derivations

There were a ring, two - algebras and a homomorphism of algebras. Then there is a natural bijection between

And

Importance for the algebraic geometry

Was for a scheme

It is a schema and a schema. The scheme is the relative tangent bundle of over. Then there is a natural bijection

For arbitrary schemas. A -valued point is thus a point -quality together with a tangent vector at that point. One can imagine a point with a tangent vector for a body so.