Differential algebra

In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natural example of a differential field is the field of rational functions C(t) in one variable, over the complex numbers, where the derivation is the differentiation with respect to t.

Differential algebra refers also to the area of mathematics consisting in the study of these algebraic objects and their use for an algebraic study of the differential equations. Differential algebra was introduced by Joseph Ritt.

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

such that each derivation ∂ satisfies the Leibniz product rule

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

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