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

Authors : Mahdi ANBARLOEİ

Pages : 171-184

Doi:10.15672/hujms.1088506

View : 14 | Download : 4

Publication Date : 2023-02-15

Article Type : Research Paper

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 $insert ignore into journalissuearticles values(m,n);$-hyperring that we call n-ary $J$-hyperideals. A proper hyperideal $Q$ of a Krasner $insert ignore into journalissuearticles values(m,n);$-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 $ginsert ignore into journalissuearticles values(x_1^n); \\in Q$ and $x_i \\notin J_{insert ignore into journalissuearticles values(m,n);}insert ignore into journalissuearticles values(R);$, then $ginsert ignore into journalissuearticles values(x_1^{i-1},1_R,x_{i+1}^n); \\in Q$. 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 $insert ignore into journalissuearticles values(k,n);$-absorbing $\\delta$-$J$-hyperideals. Let $\\delta$ be a hyperideal expansion of a Krasner $insert ignore into journalissuearticles values(m,n);$-hyperring $R$ and $k$ be a positive integer. A proper hyperideal $Q$ of $R$ is called $insert ignore into journalissuearticles values(k,n);$-absorbing $\\delta$-$J$-hyperideal if for $x_1^{kn-k+1} \\in R$, $ginsert ignore into journalissuearticles values(x_1^{kn-k+1}); \\in Q$ implies that $ginsert ignore into journalissuearticles values(x_1^{insert ignore into journalissuearticles values(k-1);n-k+2}); \\in J_{insert ignore into journalissuearticles values(m,n);}insert ignore into journalissuearticles values(R);$ or a $g$-product of $insert ignore into journalissuearticles values(k-1);n-k+2$ of $x_i^,$ s except $ginsert ignore into journalissuearticles values(x_1^{insert ignore into journalissuearticles values(k-1);n-k+2});$ is in $\\deltainsert ignore into journalissuearticles values(Q);$.
Keywords : ‎ ‎n‎ ‎ ary ‎ ‎J‎ ‎ hyperideal, ‎ ‎n‎ ‎ ary ‎‎ ‎\\delta‎ ‎ ‎ ‎J‎ ‎ hyperideal, ‎ k, n, ‎ ‎ absorbing ‎ ‎\\delta‎ ‎ ‎ ‎J‎ ‎ hyperideal