Differential algebra

A differential ring is a ring R equipped with one or more derivations, that are homomorphisms of additive groups

${\displaystyle \partial \colon R\to R\,}$

such that each derivation ∂ satisfies the Leibniz product rule

${\displaystyle \partial (r_{1}r_{2})=(\partial r_{1})r_{2}+r_{1}(\partial r_{2}),\,}$

for every ${\displaystyle r_{1},r_{2}\in R}$. Note that the ring could be noncommutative, so the somewhat standard d(xy) = xdy + ydx form of the product rule in commutative settings may be false. If ${\displaystyle M:R\times R\to R}$ is multiplication on the ring, the product rule is the identity

${\displaystyle \partial \circ M=M\circ (\partial \times \operatorname {id} )+M\circ (\operatorname {id} \times \partial ).}$

where ${\displaystyle f\times g}$ means the function which maps a pair ${\displaystyle (x,y)}$ to the pair ${\displaystyle (f(x),g(y))}$.

A differential field is a commutative field K equipped with derivations.

The well-known formula for differentiating fractions

${\displaystyle \partial \left({\frac {u}{v}}\right)={\frac {\partial (u)\,v-u\,\partial (v)}{v^{2}}}}$

follows from the product rule. Indeed, we must have

${\displaystyle \partial \left({\frac {u}{v}}\times v\right)=\partial (u)}$

By the product rule, we then have

${\displaystyle \partial \left({\frac {u}{v}}\right)\,v+{\frac {u}{v}}\,\partial (v)=\partial (u)}$

Solving with respect to ${\displaystyle \partial (u/v)}$, we obtain the sought identity.

If K is a differential field then the field of constants of K is ${\displaystyle k=\{u\in K:\partial (u)=0\}.}$

A differential algebra over a field K is a K-algebra A wherein the derivation(s) commutes with the scalar multiplication. That is, for all ${\displaystyle k\in K}$ and ${\displaystyle x\in A}$ one has

${\displaystyle \partial (kx)=k\partial x}$

If ${\displaystyle \eta \colon K\to Z(A)}$ is the ring homomorphism to the center of A defining scalar multiplication on the algebra, one has

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

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

${\displaystyle \partial (xy)=(\partial x)y+x(\partial y)}$

and

${\displaystyle \partial (ax+by)=a\,\partial x+b\,\partial y.}$

A derivation on a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is a linear map ${\displaystyle D\colon {\mathfrak {g}}\to {\mathfrak {g}}}$ satisfying the Leibniz rule:

${\displaystyle D([a,b])=[a,D(b)]+[D(a),b]}$

For any ${\displaystyle a\in {\mathfrak {g}}}$, ad(a) is a derivation on ${\displaystyle {\mathfrak {g}}}$, which follows from the Jacobi identity. Any such derivation is called an inner derivation. This derivation extends to the universal enveloping algebra of the Lie algebra.

If A is unital, then ∂(1) = 0 since ∂(1) = ∂(1 × 1) = ∂(1) + ∂(1). For example, in a differential field of characteristic zero ${\displaystyle K}$, the rationals are always a subfield of the field of constants of ${\displaystyle K}$.

Any ring is a differential ring with respect to the trivial derivation which maps any ring element to zero.

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 multiplication 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

${\displaystyle \partial (u)=u}$

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.

Naturally occurring examples of derivations are partial derivatives, Lie derivatives, the Pincherle derivative, and the commutator with respect to an element of an algebra.

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

This is the set of formal infinite sums

${\displaystyle \left\{\sum _{n\ll \infty }r_{n}\xi ^{n}\mid r_{n}\in R\right\},}$

where ${\displaystyle n\ll \infty }$ means that the sum runs on all integers that are not greater than a fixed (finite) value.

This set is made a ring with the multiplication defined by linearly extending the following formula for "monomials":

${\displaystyle (r\xi ^{m})(s\xi ^{n})=\sum _{k=0}^{\infty }r(\partial ^{k}s){m \choose k}\xi ^{m+n-k},}$

where ${\displaystyle \textstyle {m \choose k}={\frac {m(m-1)\dots (m-k+1}{k!}}}$ is the binomial coefficient. (If ${\displaystyle m>0,}$ the sum is finite, as the terms with ${\displaystyle k>m}$ are all equal to zero.) In particular, one has

${\displaystyle \xi ^{-1}s=\sum _{k=0}^{\infty }(-1)^{k}(\partial ^{k}s)\xi ^{-1-k}.}$

for r = 1, m = –1, and n = 0, and using the identity ${\displaystyle \textstyle {-1 \choose k}=(-1)^{k}}$

• 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.
• Arithmetic derivative
• Differential calculus over commutative algebras
• Difference algebra
• Differential algebraic geometry
• Picard–Vessiot theory
• Hardy field

• Buium, Alexandru (1994). Differential algebra and diophantine geometry. Hermann. ISBN 978-2-7056-6226-4.
• Kaplansky, Irving (1976). An introduction to differential algebra (2nd ed.). Hermann. ISBN 9782705612511.
• Kolchin, Ellis (1973). Differential Algebra & Algebraic Groups. Academic Press. ISBN 978-0-08-087369-5.
• Marker, David (2017) [1996]. "Model theory of differential fields". In Marker, David; Messmer, Margit; Pillay, Anand (eds.). Model Theory of Fields. Lecture notes in Logic. 5. Cambridge University Press. pp. 38–113. ISBN 978-1-107-16807-7. As PDF
• Magid, Andy R. (1994). Lectures on Differential Galois Theory. University lecture series. 7. American Mathematical Society. ISBN 978-0-8218-7004-4.

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