On 2-absorbing and 2-absorbing primary hyperideals of a multiplicative hyperring

Abstract

The δ-primary hyperideal is a concept unifing the n-ary prime and n-ary primary hyperideals under one frame where δ is a function which assigns to each hyperideal Q of G a hyperideal δ(Q) of the same hyperring with specific properties. In this paper, for a commutative Krasner (m, n)hyperring G with scalar identity 1, we aim to introduce and study the notion of (t, n)-absorbing δ-semiprimary hyperideals which is a more general structure than δ-primary hyperideals. We say that a proper hyperideal Q of G is an (t, n)-absorbing δ-semiprimary hyperideal if whenever k(a tn−t+1 1 ) ∈ Q for a tn−t+1 1 ∈ G, then there exist (t − 1)n − t + 2 of the a , i s whose k-product is in δ(Q). Furthermore, we extend the concept to weakly (t, n)-absorbing δ-semiprimary hyperideals. Several properties and characterizations of these classes of hyperideals are determined. In particular, after defining srongly weakly (t, n)-absorbing δ-semiprimary hyperideals, we present the condition in which a weakly (t, n)-absorbing δ-semiprimary hyperideal is srongly. Moreover, we show that k(Q (tn−t+1) ) = 0 where the weakly (t, n)-absorbing δsemiprimary hyperideal Q is not (t, n)-absorbing δ-semiprimary. Also, we investigate the stability of the concepts under intersection, homomorphism and cartesian product of hyperrings.2010 Mathematics Subject Classification. 16Y99.

On expansions of graded 2-absorbing hyperideals in graded multiplicative hyperrings

“…R. Ameri et al in [2] described multiplicative hyperring of fractions and coprime hyperideals. Later on, many researches have observed that generalizations of prime hyperideals in multiplicative hyperrings [5,6,8,34]. Recently, The concept of graded multiplicative hyperrings and graded hyperideals was introduced in [22].…”

On S-prime hyperideals in multiplicative hyperrings

“…Ghiasvand in [9] has introduced and studied the concept of 2-absorbing hyperideals of a multiplicative hyperring as a generalization of prime hyperideals. Also, Anbarloei has studied 2-absorbing and 2-absorbing primary hyperideals of a multiplicative hyperring in [2,3]. In this paper, we introduce and study the concept of S-prime hyperideal in a multiplicative hyperring which is also a generalization of prime hyperideals, and obtain their basic properties.…”

On phi-2-Absorbing phi‎-2-Absorbing Primary Hyperideals of A Multiplicative Hyperring

“…A proper hyperideal I of R is 2-absorbing if xoyoz ⊆ I with x, y, z ∈ R, then xoy ⊆ I or xoz ⊆ I or yoz ⊆ I. Moreover, recall from [1] that a proper hyperideal I of R is said to be 2-absorbing primary hyperideal if x, y, z ⊆ I with x, y, z ∈ R, then xoy ⊆ I or yoz ⊆ r(I) or xoz ⊆ r(I) . A proper hyperideal of R is a prime hyperideal if xoy ⊆ P with x, y ∈ R implies that x ∈ P or y ∈ P .…”