Homomorphism of Lie rings

Definition

Suppose $L_1$ and $L_2$ are Lie rings. A homomorphism of Lie rings is a map $f:L_1 \to L_2$ such that:

1. $f$ is a homomorphism of groups with respect to the additive group structures of $L_1$ and $L_2$.
2. $f([x,y]) = [f(x),f(y)]$, i.e., $f$ commutes with the Lie bracket.

In other words, a homomorphism of Lie rings is a homomorphism in the variety of Lie rings.