1-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 1-cocycle for this action is a homomorphism of groups $$f:L \to M$$ satisfying the additional condition:

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

Equivalently:

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

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

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

Definition in terms of the general definition of cocycle
A 1-cocycle for a Lie ring action is a special case of a defining ingredient::cocycle for a Lie ring action, namely $$n = 1$$.

Group structure
The set of 1-cocycles for the action of $$L$$ on $$M$$ forms a group under pointwise addition.

Related notions

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