arXiv subjects
Publications and source records attributed to Melis Bolat.
In this study, we introduce graded pseudo weakly prime submodules of G-graded R-modules, which are an extension of graded weakly prime ideals over G-graded rings. On the graded spectrum of graded pseudo weakly prime submodules, we investigate the Zariski topology. Different aspects of this topological space are investigated, and they are linked to the algebraic properties of the G-graded R-modules under study.
In this paper, our purpose is to define the expansion of $r$-hyperideals and extend this concept to $ϕ$-$δ$-$r$-hyperideal. Let $\Re$ be a commutative Krasner hyperring with nonzero identity. Given an expansion $δ$ of hyperideals, a proper hyperideal $N$ of $\Re$ is called $δ$-$r$-hyperideal if $a\cdot b\in N$ with $ann(a)=0$ implies that $b\in δ(N)$, for all $a,b\in\Re$. Therefore, given an expansion $δ$ of hyperideals and a hyperideal reduction $ϕ$, a proper hyperideal $N$ of $\Re$ is called $ϕ$-$δ$-$r$-hyperideal if $a\cdot b\in N-ϕ(N)$ with $ann(a)=0$ implies that $b\inδ(N)$, for all $a,b\in\Re$. We investigate some of their properties and give some examples.
In this study, we examine some properties of $r$-hyperideals in the commutative Krasner hyperrings. Some properties of $pr$-hyperideals are also studied. The relation between prime hyperideals and $r$-hyperideals is investigated. We show that the image and the inverse image of an $r$-hyperideal is also an $r$-hyperideal. We also introduce a generalization of r-hyperideals and we prove some properties of them.
In this paper, we study commutative Krasner hyperring with nonzero identity. $ϕ$-prime, $ϕ$-primary and $ϕ$-$δ$-primary hyperideals are introduced. We intend to extend the concept of $δ$-primary hyperideals to $ϕ$-$δ$-primary hyperideals. We give some characterizations of hyperideals to classify them. We denote the set of all hyperideals of $\Re$ by $L(\Re)$ (all proper hyperideals of $\Re$ by $L^{\ast }(\Re)).$ Let $ϕ$ be a reduction function such that $ϕ:L(\Re)\rightarrow L(\Re)\cup\{\emptyset\}$ and $δ$ be an expansion function such that $δ:L(\Re)\rightarrow L(\Re).$ $N$ be a proper hyperideal of $\Re.$ $N$ is called $ϕ$-$δ$-primary hyperideal of $\Re$ if $a\circ b\in N-$ $ϕ(N),$ then $a\in N$ or $b\inδ(N),$ for some $a,b\in\Re.$ We\ discuss the relation between $ϕ$-$δ$-primary hyperideal and other hyperideals.