$\varphi$-$(k,n)$-absorbing and $\varphi$-$(k,n)$-absorbing primary hyperideals in a krasner $(m,n)$-hyperring

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 $Q$ $g(r_1^{(k-1)n-k+2})\in Q$ 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 ${\mathbf{r}}^{(\mathbf{m},\mathbf{n})}(Q)$. In this paper, we aim to extend the notions to the concepts of $\varphi$-$(k,n)$-absorbing and $\varphi$-$(k,n)$-absorbing primary hyperideals. Assume that $\varphi$ is a function from $\mathcal{H}\mathcal{I}(R)$ to $\mathcal{H}\mathcal{I}(R)\cup \{\varnothing \}$ such that $\mathcal{H}\mathcal{I}(R)$ is the set of hyperideals of $R$ and $k$ is a positive integer. We call a proper hyperideal $Q$ of $R$ a $\varphi$-$(k,n)$-absorbing primary hyperideal if for $r_1^{kn-k+1}\in R$, $g(r_1^{kn-k+1})\in Q-\varphi (Q)$ implies that there are $(k-1)n-k+2$ of the $r_i^{,}$s whose $g$-product is in $Q$ $g(r_1^{(k-1)n-k+2})\in Q$ 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 ${\mathbf{r}}^{(\mathbf{m},\mathbf{n})}(Q)$. Several properties and characterizations of them are presented.