Hypercomplex numbers of the form
,
where
and
are real numbers, and where the
double numbers
satisfy the relation
,
while the
dual numbers
satisfy the relation
(cf.
Hypercomplex number).
Addition of double and dual numbers is defined by
Multiplication of double numbers is defined by
and that of dual numbers by
Complex numbers, double numbers and dual numbers are also called
complex numbers of hyperbolic,
elliptic
and
parabolic types,
respectively. These numbers are sometimes used to represent motions
in the three-dimensional spaces of Lobachevskii,
Riemann and
Euclid
(see, for instance,
Helical calculus).
Both double and dual numbers form two-dimensional (with base 1 and
)
associative-commutative algebras over the field of real numbers. As
distinct from the field of complex numbers, these algebras
comprise zero divisors, all these having the form
in the algebra of double numbers. The algebra of double numbers may be
split into a direct sum of two real number
fields. Hence yet another name for double numbers —
splitting complex numbers.
Double numbers have yet another appellation —
paracomplex numbers.
The algebra of dual numbers is considered not only over the field
of real numbers, but also over an arbitrary field or commutative ring. Let
be a commutative ring and let
be an
-module.
The direct sum of
-modules
equipped with the multiplication
is a commutative
-algebra
and is denoted by
.
It is known as the
algebra of dual numbers with respect to the module
.
The
-module
is identical with the ideal of the algebra
which is the kernel of the augmentation homomorphism
The square
of this ideal is zero, while
.
If
is a regular ring the converse is also true: If
is an
-algebra
and
is an ideal in
such that
and
,
then
,
where
is regarded as an
-module
[4].
If
,
the algebra
(then denoted by
)
is isomorphic to the quotient algebra of the algebra of polynomials
by the ideal
.
Many properties of an
-module
may be formulated as properties of the algebra
;
as a result, many problems on
-modules
can be reduced to corresponding problems in the theory of rings
[2].
Let
be an arbitrary
-algebra,
let
be a homomorphism and let
be a derivation (cf.
Derivation in a ring)
of
with values in the
-module
,
regarded as a
-module
with respect to the homomorphism
.
The mapping
(
)
will then be a homomorphism of
-algebras.
Conversely, for any homomorphism of
-algebras
the composition
,
where
is the projection of
onto
,
is an
-derivation
of
with values in
,
regarded as a
-module
with respect to the homomorphism
.
This property of double and dual numbers is utilized for
the description of the tangent space to an arbitrary functor in the category of schemes
[1],
[3].