Several researchers like Armendariz, Efraim, Dunget al, Page, S.S, Yousif, Fuller, Anderson have been studiedchain conditions on essential ideals in modules. We extend the conceptof ascending chain condition to Near-ring groups defining weaklyNoetherianN-group. Introducing almost weaklyNoetherian N-group we established that for dgnr N, N-group E is almost weaklyNoetherian, E hasA.C.C. on ... [Show full abstract] essential ideals and E/M is weaklyNoetherianfor every essential ideal M of E are equivalent. These relations motivates us to study the relation between weaklyNoetherian Ngroup E and A.C.C. on essential ideals of E. Finally we have shown Near-ring N is weakly Noetherian if ⊕i∈IEi of injective N-groups is injective.