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

Abstract

‎Let $R$ be a multiplicative hyperring‎. ‎In this paper‎, ‎we extend the concept of 2-absorbing hyperideals and 2-absorbing primary hyperideals to the context ‎$‎\varphi‎$‎-2-absorbing hyperideals and ‎$‎\varphi‎$‎-2-absorbing primary hyperideals. Let ‎$‎E(R)‎$‎ be the set of hyperideals of ‎$‎R‎$‎‎ and ‎$\varphi : E(R) \longrightarrow E(R) \cup \{\phi\}‎$‎ be a function. A nonzero proper hyperideal ‎$‎I‎$‎ of ‎$‎R‎$‎ is called a ‎$\varphi‎$‎- 2-absorbing hyperideal if for all ‎$x, y, z \in R, xoyoz \subseteq I- \varphi(I)‎$‎ implies‎$xoy \subseteq I‎$‎ or ‎$‎yoz \subseteq I‎$‎ or ‎$‎xoz \subseteq I‎$‎. Also, a nonzero proper hyperideal ‎$‎I‎$‎ of ‎$‎R‎$‎ is called a ‎$\varphi‎$‎- 2-absorbing primary hyperideal if for all ‎$x, y, z \in R, \ xoyoz \subseteq I- \varphi(I)‎$‎ implies‎$xoy \subseteq I‎$‎ or ‎$‎yoz \subseteq r(I)‎$‎ or ‎$‎xoz \subseteq r(I)‎$‎. A number of results concerning them are given.

$J$-hyperideals and their expansions in a Krasner $(m,n)$-hyperring

“…1 ) ∈ P for some r n 1 ∈ R. Since P ⊆ r (m,n) (P ) and P is an n-ary q-primary hyperideal of R, we get r i ∈ r (m,n) (P ) for some 1 ≤ i ≤ n. Since g(r (m,n) (P ) (2) , 1 (n−2) ) ⊆ P , we have g(r…”

“…Definition 2.1. A proper hyperideal P of R is called n-ary quasi-primary (briefly, q-primary) provided that r (m,n) (P ) is an n-ary prime hyperideal of R. Consider the hyperideal P = { 0, 4} of G. Then we have r (2,2) (P ) = { 0, 2, 4, 6, }. It is easy to see that the radical of the hyperideal P is prime and so P is q-primary.…”

“…and for a 2 1 ∈ H, g(0, a 2 1 ) = 0. Then the hyperideal T = {0, 2} is a 3-ary sqprimary hyperideal of H. Theorem 3.3.…”

“…Hence r (m,n) (P ) is an n-ary prime hyperideal of R. Thus P is an n-ary q-primary hyperideal of R. Now, we determine when an n-ary q-primary hyperideal of R is an n-ary sqprimary hyperideal of R. Theorem 3.4. Let P be a proper hyperideal of R such that g(r (m,n) (P ) (2) , 1 (n−2) ) ⊆ P . If P is an n-ary q-primary hyperideal of R, then P is an n-ary sq-primary hyperideal of R.…”

“…Theorem 3.6. Let P be a proper hyperideal of R. If P is an n-ary sq-primary hyperideal of R, then g(P n 1 ) ⊆ P for some hyperideals P n 1 of R implies that g(p (2) , 1 (n−2) ) ∈ P for all p ∈ P i or g(P i−1…”