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 in one variable over the complex numbers, where the derivation is differentiation with respect to Differential algebra refers also to the area of mathematics consisting in the study of these algebraic objects and their use in the algebraic study of differential equations. Differential algebra was introduced by Joseph Ritt in 1950. Open problems The biggest open problems in the field include the Kolchin catenary conjecture, the Ritt problem, and the Jacobi bound problem. All of these deal with the structure of differential ideals in differential rings. Differential ring A differential ring is a ring equipped with one or more derivations, which are homomorphisms of additive groups such that each derivation satisfies the Leibniz product rule for every Note that the ring could be noncommutative, so the somewhat standard form of the product rule in commutative settings may be false. If is multiplication on the ring, the product rule is the identity where means the function which maps a pair to the pair Note that a differential ring is a (not necessarily graded) -differential algebra. Differential field A differential field is a commutative field equipped with derivations. The well-known formula for differentiating fractions follows from the product rule. Indeed, we must have By the product rule, Solving with respect to we obtain the sought identity. If is a differential field then the field of constants of is A differential algebra over a field is a -algebra wherein the derivation(s) commutes with the scalar multiplication. That is, for all and If is the ring homomorphism to the center of A defining scalar multiplication on the algebra, one has As above, the derivation must obey the Leibniz rule over the algebra multiplication, and must be linear over addition. Thus, for all and and Derivation on a Lie algebra A derivation on a Lie algebra is a linear map satisfying the Leibniz rule: For any is a derivation on 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. Examples If is a unital algebra, then since For example, in a differential field