Differential algebra

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.

Differential ring

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

\partial:R \to R

such that each derivation satisfies the Leibniz product rule

\partial(r_1 r_2)=(\partial r_1) r_2 + r_1 (\partial r_2).\,

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

\partial \circ M =

M \circ (\partial \times \operatorname{Id}) + M \circ (\operatorname{Id} \times \partial)

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

\partial(uv) = u \partial v + v \partial u

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

\partial (u + v) = \partial u + \partial v\,

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

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

k \in K

and

x \in A

one has

\partial (kx) = k \partial x

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

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

M \circ (\eta \times \partial)

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

\partial (xy) = (\partial x) y + x(\partial y)

and

\partial (ax+by) = a\partial x + b\partial y

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

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

Multiplication on this ring is defined as

(r\xi^m)(s\xi^n) =

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

Here ${m \choose k}$ is the binomial coefficient. Note the identities

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

which makes use of the identity

{-1 \choose n} = (-1)^n

and

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

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Differential ring

Differential field

Differential algebra

Ring of pseudo-differential operators

See also