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)
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)