Merging the N-hyperideals and J-hyperideals in one frame

The notions of [Formula: see text]-hyperideals and [Formula: see text]-hyperideals as two new classes of hyperideals were recently defined in the context of Krasner [Formula: see text]-hyperrings. These concepts are created on the basis of the intersection of all [Formula: see text]-ary prime hyperideals and the intersection of all maximal hyperideals, respectively. Despite being vastly different in many aspects, they share numerous similar properties. The aim of this research work is to merge them under one frame called [Formula: see text]-ary [Formula: see text]-hyperideals where the function [Formula: see text] assigns to each hyperideal of a Krasner [Formula: see text]-hyperring a hyperideal of the same hyperring. We give various properties of [Formula: see text]-ary [Formula: see text]-hyperideals and use them to characterize certain classes of hyperrings such as hyperintegral domains and local hyperrings. Moreover, we introduce the notion of weakly [Formula: see text]-absorbing [Formula: see text]-hypereideals and present a copy of Nakayama’s lemma for a type of weakly [Formula: see text]-absorbing [Formula: see text]-hypereideals.