$\phi$-$(k,n)$-absorbing (primary) hyperideals in a Krasner $(m,n)$-hyperring

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran.

Abstract

Various expansions of prime hyperideals have been studied in a Krasner $(m,n)$-hyperring $R$. For instance, a proper hyperideal $Q$ of $R$ is called weakly $(k,n)$-absorbing (primary) provided that for $r_1^{kn-k+1} \in R$, $g(r_1^{kn-k+1}) \in Q-\{0\}$ implies that there are $(k-1)n-k+2$ of the $r_i^,$s whose $g$-product is in $Q$ $\Bigl ($ $g(r_1^{(k-1)n-k+2}) \in Q$ or a $g$-product of $(k-1)n-k+2$ of $r_i^,$s ,except $g(r_1^{(k-1)n-k+2})$, is in $\boldsymbol{ r}^{(m,n)}(Q)$ $\Bigr )$. In this paper, we aim to extend the notions to the concepts of $\phi$-$(k,n)$-absorbing and $\phi$-$(k,n)$-absorbing primary hyperideals. Assume that $\phi$ is a function from $ \mathcal{HI}(R)$ to $\mathcal{HI}(R) \cup \{\varnothing\}$ such that $\mathcal{HI}(R)$ is the set of hyperideals of $R$ and $k$ is a positive integer. We call a proper hyperideal $Q$ of $R$ a $\phi$-$(k,n)$-absorbing (primary) hyperideal if for $r_1^{kn-k+1} \in R$, $g(r_1^{kn-k+1}) \in Q-\phi(Q)$ implies that there are $(k-1)n-k+2$ of the $r_i^,$s whose $g$-product is in $Q$ $\Bigl ($ $g(r_1^{(k-1)n-k+2}) \in Q$ or a $g$-product of $(k-1)n-k+2$ of $r_i^,$s ,except $g(r_1^{(k-1)n-k+2})$, is in $\boldsymbol{ r}^{(m,n)}(Q)$ $\Bigr )$. Several properties and characterizations of them are presented.

Keywords

References

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Algebraic Structures and Their Applications
Volume 11, Issue 4
November 2024
Pages 287-304

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