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

Abstract

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 $\delta(Q)$.

Keywords

• ‎$‎n‎$‎-ary ‎$‎J‎$‎-hyperideal
• ‎$‎n‎$‎-ary ‎‎$‎\delta‎$‎-‎$‎J‎$‎-hyperideal
• $‎(k,n)‎$‎-absorbing ‎$‎\delta‎$‎-‎$‎J‎$‎-hyperideal

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