Properties
The Leibniz law itself has a number of immediate consequences. Firstly, if x1, x2, … ,xn ∈ A, then it follows by mathematical induction that
In particular, if A is commutative and x1 = x2 = … = xn, then this formula simplifies to the familiar power rule D(xn) = nxn−1D(x). If A is unital, then D(1) = 0 since D(1) = D(1·1) = D(1) + D(1). Thus, since D is K linear, it follows that D(x) = 0 for all x ∈ K.
If k ⊂ K is a subring, and A is a k-algebra, then there is an inclusion
since any K-derivation is a fortiori a k-derivation.
The set of k-derivations from A to M, Derk(A,M) is a module over k. Furthemore, the k-module Derk(A) forms a Lie algebra with Lie bracket defined by the commutator:
It is readily verified that the Lie bracket of two derivations is again a derivation.
Read more about this topic: Derivation (abstract Algebra)
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