In mathematics, differential rings, differential fields and differential algebras are rings, fields and algebras equipped with a derivation, which is a unary function satisfying the Leibniz product law. A natural example of a differential field is the field of rational functions over the complex numbers in one variable, C(t), where the derivation is differentiation with respect to t. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

Differential ring

A differential ring is a ring R equipped with one or more derivations

such that each derivation satisfies the Leibniz product rule In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

for every . Note that the ring may not be commutative, so the somewhat standard d(xy) = xdy + ydx form of the product rule in commutative settings may be false. In index-free notation, if is multiplication on the ring, the product rule is the identity

where means the function which maps a couple (x,y) to the couple (f(x),g(y)).

Differential field

A differentials field is a field K, together with a derivation. The theory of differential fields, DF, is given by the usual field axioms along with two extra axioms involving the derivation. As above, the derivation must obey the product rule, or Leibniz rule over the elements of the field, in order to be worthy of being called a derivation. That is, for any two elements u, v of the field, one has In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

since multiplication on the field is commutative. The derivation must also be distributive over addition in the field:

If K is a differential field then the field of constants

Differential algebra

A differential algebra over a field K is a K-algebra A wherein the derivation(s) commutes with the field. That is, for all and one has

In index-free notation, if is the ring morphism defining scalar multiplication on the algebra, one has In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

As above, the derivation must obey the Leibniz rule over the algebra multiplication, and must be linear over addition. Thus, for all and one has

and

Examples

If A is unital, then ∂(1) = 0 since ∂(1) = ∂(1 × 1) = ∂(1) + ∂(1). For example, in a differential field of charecteristic zero the rationals are always a subfield of the constant field. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ...

Any field pure can be interpretted as a constant differential field.

The field Q(t) has a unique structure as a differential field, determined by setting ∂(t) = 1: the field axioms along with the axioms for derivations ensure that the derivation is differentiation with respect to t. For example, by commutativity of addition and the Leibniz law one has that ∂(u²) = u ∂(u) + ∂(u)u= 2u∂(u).

The differential field Q(t) fails to have a solution to the differential equation

but expands to a larger differential field including the function e^t 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. This article is considered orphaned, since there are very few or no other articles that link to this one. ... // Motivation and basic idea In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. ...

Naturally occurring examples of derivations are partial derivatives, Lie derivatives, the Pincherle derivative, and the commutator with respect to an element of the algebra. All these examples are tightly related, with the concept of derivation as the major unifying theme. 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). ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... In mathematics, the Pincherle derivative of a linear operator T on the space of polynomials in x is another linear operator T′ defined by which means that for any polynomial f(x), This is a derivation satisfying the sum and product rules: (T + S)′ = T′ + S&#8242... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

Ring of pseudo-differential operators

Differential rings and differential algebras are often studied by means of the ring of pseudo-differential operators on them. In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. ...

This is the ring

Multiplication on this ring is defined as

Here is the binomial coefficient. Note the identities In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ...

which makes use of the identity

and

Graded derivations

If we have a graded algebra A, and D is an homogeneous linear map of grade on A then D is an homogeneous derivation if D(ab) = D(a)b + ε ^{| a | | D |} aD(b), acting on homogeneous elements of A. A graded derivation is sum of homogeneous derivations with the same ε. In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ...

If the commutator factor ε = 1, this definition reduces to the usual case.

If ε = − 1, however, we have D(ab) = D(a)b + ( − 1) ^{| a |} aD(b), for odd . They are called antiderivations.

Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the interior product is a degree −1 derivation on the exterior algebra of differential forms on a smooth manifold. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

Graded derivations of superalgebras (i.e.-graded algebras) are often called superderivations. In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2). ...

See also

• Differential Galois theory
• Kähler differential
• Differentially closed field
• A D-module is an algebraic structure with several differential operators acting on it.
• A differential graded algebra is a differential algebra with an additional grading.

// Motivation and basic idea In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. ... In mathematics, the Kähler differentials are a universal construction Ω1S/R associated to a ring homomorphism of commutative rings, φ:R → S, that provides an analogue of the construction of differential forms (1-forms). ... This article is considered orphaned, since there are very few or no other articles that link to this one. ... In mathematics, a D-module is a module over a ring D of differential operators. ... In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure. ...

References

• Buium, Differential Algebra and Diophantine Geometry, Hermann (1994).
• I. Kaplansky, Differential Algebra, Hermann (1957).
• E. Kolchin, Differential Algebra and Algebraic Groups, 1973
• D. Marker, Model theory of differential fields, Model theory of fields, Lecture notes in Logic 5, D. Marker, M. Messmer and A. Pillay, Springer Verlang (1996).
• A. Magid, Lectures on Differential Galois Theory, American Math. Soc., 1994

External links

• David Marker's home page has several online surveys discussing differential fields.