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Differential algebra

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In mathematics, in the area of ring theory, differential rings, differential fields and differential algebras are rings, fields and algebras equipped with a derivation. The definitions of each are closely related and are all presented here.

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[edit] Differential ring

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

<math>\partial:R \to R</math>

such that each derivation satisfies the Leibniz product rule

<math>\partial(r_1 r_2)=(\partial r_1) r_2 + r_1 (\partial r_2).\,</math>

for every <math>r_1, r_2 \in R</math>. In index-free notation, if <math>M:R \times R \to R</math> is multiplication on the ring, the product rule is the identity

<math>\partial \circ M =

M \circ (\partial \times \operatorname{Id}) + M \circ (\operatorname{Id} \times \partial) </math>

[edit] Differential field

A differential field is a field F, together with a derivation. As above, the derivation must obey the 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

<math>\partial(uv) = u \partial v + v \partial u</math>

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

<math>\partial (u + v) = \partial u + \partial v\, </math>

Differential fields are the object of study in differential Galois theory.

[edit] 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 <math>k \in K</math> and <math>x \in A</math> one has

<math>\partial (kx) = k \partial x</math>

In index-free notation, if <math>\eta:K\to A</math> is the ring morphism defining scalar multiplication on the algebra, one has

<math>\partial \circ M \circ (\eta \times \operatorname{Id}) =

M \circ (\eta \times \partial)</math>

As above, the derivation must obey the Leibniz rule over the algebra multiplication, and must be linear over addition. Thus, for all <math>a,b \in K</math> and <math>x,y \in A</math> one has

<math>\partial (xy) = (\partial x) y + x(\partial y)</math>

and

<math>\partial (ax+by) = a\partial x + b\partial y</math>

[edit] Ring of pseudo-differential operators

Differential rings and differential algebras are often studied by means of the ring of pseudo-differential operators on them.

This is the ring

<math>R((\xi^{-1})) = \left\{ \sum_{n<\infty} r_n \xi^n | r_n \in R \right\}</math>

Multiplication on this ring is defined as

<math>(r\xi^m)(s\xi^n) =

\sum_{k=0}^m r (\partial^k s) {m \choose k} \xi^{m+n-k}</math>

Here <math>{m \choose k}</math> is the binomial coefficient. Note the identities

<math>\xi^{-1} r = \sum_{n=0}^\infty (-1)^n (\partial^n r) \xi^{-1-n}</math>

which makes use of the identity

<math>{-1 \choose n} = (-1)^n</math>

and

<math>r \xi^{-1} = \sum_{n=0}^\infty \xi^{-1-n} (\partial^n r).</math>

[edit] See also

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