Exterior product of Lie rings

For Lie rings that are ideals in a big ring
Suppose $$M,N$$ are (possibly equal, possibly distinct) ideals in a Lie ring $$Q$$ (Note that in fact it suffices to assume that they idealize each other, but there is no loss of generality in assuming that they are both ideals). Define a compatible pair of actions of Lie rings of $$M$$ and $$N$$ on each other via the adjoint action on each other, i.e., the action that each induces on the other by restricting the inner derivation given by the adjoint action in the whole group (see Lie ring acts as derivations by adjoint action). The exterior product of $$M$$ and $$N$$ is then defined as the quotient of the tensor product of Lie rings $$M \otimes N$$ for this compatible pair of actions by the ideal generated by elements of the form $$x \otimes x, x \in M \cap N$$.