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More generally, if M is an A-bimodule, a K-linear map that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by .
since it is readily verified that the commutator of two derivations is again a derivation.
There is an A-module ?A/K (called the Kähler differentials) with a K-derivation d: A -> ?A/K through which any derivation D: A -> M factors. That is, for any derivation D there is a A-module map ? with
The correspondence is an isomorphism of A-modules:
If k ? K is a subring, then A inherits a k-algebra structure, so there is an inclusion
since any K-derivation is a fortiori a k-derivation.
Given a graded algebraA and a homogeneous linear map D of grade || on A, D is a homogeneous derivation if
for every homogeneous element a and every element b of A for a commutator factor . A graded derivation is sum of homogeneous derivations with the same ?.
If , this definition reduces to the usual case. If , however, then