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.