2-cocycle for a Lie ring action

Definition
Suppose $$L$$ is a Lie ring, $$M$$ is an abelian group, and $$\varphi:L \to \operatorname{End}(M)$$ is a defining ingredient::Lie ring homomorphism from $$L$$ to the ring of endomorphisms of $$M$$, where the latter gets the usual Lie ring structure from its structure as an associative ring.

Explicit definition
A 2-cocycle for this action is a homomorphism of groups $$f:L \times L \to M$$ (where $$L \times L$$ is the external direct product) satisfying the additional condition:

$$\! \varphi(x)(f(y,z)) - \varphi(y)(f(x,z)) + \varphi(z)(f(x,y)) - f([x,y],z) - f(y,[x,z]) + f(x,[y,z]) = 0\forall \ x,y,z \in L$$

If we suppress $$\varphi$$ and denote the action by $$\cdot$$, this can be written as:

$$\! x \cdot f(y,z) - y \cdot f(x,z) + z \cdot f(x,y) - f([x,y],z) - f(y,[x,z]) + f(x,[y,z]) = 0\forall \ x,y,z \in L$$

Definition in terms of the general definition of cocycle
A 2-cocycle for a Lie ring action is a special case of a defining ingredient::cocycle for a Lie ring action, namely $$n = 2$$.

Group structure
The set of 2-cocycles for the action of $$L$$ on $$M$$ forms a group under pointwise addition.

Related notions

 * 1-cocycle for a Lie ring action
 * 2-cocycle for a group action
 * 1-cocycle for a group action