Relative essential ideals in N-groups

Let G be an N-group where N is a (right) nearring. We introduce the concept of relative essential ideal (or N-subgroup) as a generalization of the concept of an essential submodule of a module over a ring or a nearring. We provide suitable examples to distinguish between the notions relative essential and essential ideals. We prove the important properties and obtain equivalent conditions for the relative essential ideals (or N-subgroups) involving the quotient. Further, we derive results on direct sums, complement ideals of N-groups, and obtain their properties under homomorphism.