Definition:Congruence Modulo Subgroup/Additive Group of Ring

Comment on Congruence Modulo Subgroup
Let $G$ be a group, and let $H$ be a subgroup of $G$. Some authors introduce the concept of congruence modulo $H$ in the context of ring theory.

In this case, the group $G$ is taken to be the additive group of a ring.

This is acceptable, but such a treatment does presuppose that $G$ is abelian.

In such a context, all the richness of the analysis of normal subgroups is disappointingly bypassed.