In Algebraic Geometry over Groups, Baumslag, Myasnikov, and Remeslennikov define a $G$-group to be a group $H$ together with a group homomorphism $G \hookrightarrow H$. Then they define a zero-divisor in $H$ to be an element $x \in H$ for which there is some $y$ satisfying $$[x, y^g] = 1 \qquad \text{ for all } g \in G$$
See for instance the beginning of Section 1.3 on page 4 here.
Can someone say something about the motivation for this definition, and how the analogy to the ring-theoretic case works? Previously they've defined an $H$-module as a homomorphism of $G$-groups $H \to {\rm Aut}(A)$, so at first it appeared that the analogy here might be that we're taking only the multiplicative structure of a commutative ring. But that doesn't seem to carry across here.
(My tags here are sort of a random guess. Feel free to improve.)