2-cocycle for trivial Lie ring action

Definition

Suppose $L$ is a Lie ring and $M$ is an abelian group.

Explicit definition

A 2-cocycle for trivial Lie ring action is a homomorphism of groups $f: L \times L \to M$ (where $L \times L$ is the external direct product of $L$ with itself) satisfying the following additional condition:

$\! f(x,[y,z]) = f([x,y],z) + f(y,[x,z]) \ \forall \ x,y,z \in L$

Definition in terms of 2-cocycle

A 2-coycle for trivial Lie ring action is a special case of a 2-cocycle for a Lie ring action in the case where the action is trivial.