Unifing the prime and primary hyperideals under one frame in a Krasner (m, n)-hyperring

Abstract

The notions of N -hyperideals and J-hyperideals as two new classes of hyperideals were recently defined in the context of Krasner (m, n)-hyperrings. These concepts are created on the basis of the intersection of all n-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 n-ary δ(0)-hyperideals where the function δ assigns to each hyperideal of a Krasner (m, n)-hyperring a hyperideal of the same hyperring. We give various properties of n-ary δ(0)-hyperideals and use them to characterize certain classes of hyperrings such as hyperintegral domains and local hyperrings. Moreover, we introduce the notions of (s, n)-absorbing δ(0)-hypereideals and weakly (s, n)absorbing δ(0)-hypereideals.

Extensions of $n$-ary prime hyperideals via an $n$-ary multiplicative subset in a Krasner $(m,n)$-hyperring

“…follows that there exists t ∈ N such that g(g(s (3) , 1 (n−3) ) (t) , 1 (n−t) ) ∈ I for t ≤ n or g (l) (g(s (3) , 1 (n−3) ) (t) ) ∈ I for t = l(n − 1) + 1. In both possibilities, we conclude that I ∩ S = ∅ which is a contradiction.…”

“…Ozel Ay et al generalized the notion of δ-primary on Krasner hyperrings [30]. The concept of δ-primary hyperideals in Krasner (m, n)-hyperrings, which unifies the prime and primary hyperideals under one frame, was defined in [3]. In this paper, we aim to complete this circle of ideas.…”

Krasner (m,n)-hyperring of fractions

“…The notion of n-ary 2-absorbing hyperideals in a Krasner (m, n)-hyperring as a generalization of the n-ary prime hyperideals was introduced in [3]. To unify the concepts of the prime and primary hyperideals under one frame, the notion of δ-primary hyperideals was defined in Krasner (m, n)-hyperrings [4]. The formation of rings of fractions and the associated process of localization are the most important technical tools in commutative algebra.…”

2-Prime Hyperideals

“…In [35], Ulucak introduced the concepts δ-primary and 2absorbing δ-primary hyperideal over multiplicative hyperrings. In [7], we investigate δ-primary hyperideals in a Krasner (m, n)-hyperring which unify prime hyperideals and primary hyperideals.…”