In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, differential rings, differential fields, and differential algebras are
rings,
fields, and
algebras equipped with finitely many
derivations, which are
unary functions that are
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and satisfy the
Leibniz product rule. A natural example of a differential field is the field of
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s in one variable over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s,
where the derivation is differentiation with respect to
Differential algebra refers also to the area of mathematics consisting in the study of these algebraic objects and their use in the algebraic study of differential equations. Differential algebra was introduced by
Joseph Ritt in 1950.
Open problems
The biggest open problems in the field include the
Kolchin Catenary Conjecture, the
Ritt Problem
Ritt is a given name and a surname. Notable people with the name include:
*Joseph Ritt (1893–1951), American mathematician at Columbia University
*Martin Ritt (1914–1990), American director, actor, and playwright in both film and theater
*Rit ...
, and
The Jacobi Bound Problem.
All of these deal with the structure of differential ideals in differential rings.
Differential ring
A ''differential ring'' is a ring
equipped with one or more ''
derivations'', which are
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s of
additive groups
such that each derivation
satisfies the
Leibniz product rule
for every
Note that the ring could be noncommutative, so the somewhat standard
form of the product rule in commutative settings may be false. If
is multiplication on the ring, the product rule is the identity
where
means the function which maps a pair
to the pair
Note that a differential ring is a (not necessarily graded)
-differential algebra.
Differential field
A differential field is a commutative field
equipped with derivations.
The well-known formula for differentiating fractions
follows from the product rule. Indeed, we must have
By the product rule,
Solving with respect to
we obtain the sought identity.
If
is a differential field then ''the field of constants'' of
is
A differential algebra over a field
is a
-algebra
wherein the derivation(s) commutes with the scalar multiplication. That is, for all
and
If
is the
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
to the
center of A defining
scalar multiplication on the algebra, one has
As above, the derivation must obey the Leibniz rule over the algebra multiplication, and must be linear over addition. Thus, for all
and
and
Derivation on a Lie algebra
A derivation on a
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 ...
is a linear map
satisfying the Leibniz rule:
For any
is a derivation on
which follows from the
Jacobi identity. Any such derivation is called an inner derivation. This derivation extends to the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
of the Lie algebra.
Examples
If
is a
unital algebra
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
, then
since
For example, in a differential field of characteristic zero
the rationals are always a subfield of the field of constants of
.
Any ring is a differential ring with respect to the trivial derivation which maps any ring element to zero.
The field
has a unique structure as a differential field, determined by setting
the field axioms along with the axioms for derivations ensure that the derivation is differentiation with respect to
For example, by commutativity of multiplication and the Leibniz law one has that
The differential field
fails to have a solution to the differential equation
but expands to a larger differential field including the function
which does have a solution to this equation.
A differential field with solutions to all systems of differential equations is called a
differentially closed field. Such fields exist, although they do not appear as natural algebraic or geometric objects. All differential fields (of bounded cardinality) embed into a large differentially closed field. Differential fields are the objects of study in
differential Galois theory.
Naturally occurring examples of derivations are
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s,
Lie derivatives, the
Pincherle derivative, and the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
with respect to an element of an
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 ...
.
Weyl Algebra
Every differential ring
has a naturally associated
Weyl Algebra which is a noncommutative ring where
and
satisfy the relation
.
Such
modules are called
D-modules
In mathematics, a ''D''-module is a module over a ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has ...
.
In particular
itself is a