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.
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.
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.
Over the years, different types of hyperideals have been introduced in order to let us fully realize the structures of hyperrings in general.
The aim of this research work is to define and characterize a new class of hyperideals in a Krasner $(m,n)$-hyperring that we call n-ary $J$-hyperideals. A proper hyperideal $Q$ of a Krasner $(m,n)$-hyperring with the scalar identity $1_R$ is said to be an n-ary $J$-hyperideal if whenever $x_1^n \in R$ such that $g(x_1^n) \in Q$ and $x_i \notin J_{(m,n)}(R)$, then $g(x_1^{i-1},1_R,x_{i+1}^n) \in Q$. Also, we study the concept of n-ary $\delta$-$J$-hyperideals as an expansion of n-ary $J$-hyperideals. Finally, we extend the notion of n-ary $\delta$-$J$-hyperideals to $(k,n)$-absorbing $\delta$-$J$-hyperideals. Let $\delta$ be a hyperideal expansion of a Krasner $(m,n)$-hyperring $R$ and $k$ be a positive integer. A proper hyperideal $Q$ of $R$ is called $(k,n)$-absorbing $\delta$-$J$-hyperideal if for $x_1^{kn-k+1} \in R$, $g(x_1^{kn-k+1}) \in Q$ implies that $g(x_1^{(k-1)n-k+2}) \in J_{(m,n)}(R)$ or a $g$-product of $(k-1)n-k+2$ of $x_i^,$ s except $g(x_1^{(k-1)n-k+2})$ is in $\delta(Q)$.
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