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.

1 Differential ring
2 Differential field
3 Differential algebra
4 Ring of pseudo-differential operators
5 See also

[edit] 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

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

[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

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

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.

[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 $k \in K$ 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