Differential Algebra - Differential Ring

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

such that each derivation ∂ satisfies the Leibniz product rule

for every . 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 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).

where means the function which maps a pair to the pair .

Read more about this topic:  Differential Algebra

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