2-cocycle for trivial Lie ring action

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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.