In this paper, the notions of integral closure of hyperrings and hyperideals in a Krasner hyperring $(R, +, \cdot)$ are defined and some basics properties of them are studied. We define also the notion of hypervaluation hyperideals and then a relations between hypervaluations, integral closure of hyperideals and primary hyperideals are studied. In fact it is shown that the integral closure of a hyperideal is determined by the hypervaluation Krasner hyperrings.
Krasner F^{(m,n)}-hyperring were introduced and investigated by Farshi and Davvaz. In this paper, our purpose is to define and characterize three classes of F-hyperideals in a Krasner F^{(m,n)}-hyperring, namely, prime F-hyperideals, maximal F-hyperideals and primary F-hyperideals.
Primary hyperideals have been introduced and studied in multiplicative hyperrings. In this paper, we intend to study extensively primary hyperideals of multiplicative hyperrings with absorbing zero and prove some results regarding them. Also, we describe Cu- ideals of multiplicative hyperrings which are particular classes of hyperideals. In the last section, we introduce 2-absorbing primary hyperideals and investigate the properties of this notion in commutative multiplicative hyperrings.
Let R be a Krasner (m,n)-hyperring and S be an n-ary multiplicative subset of R. The purpose of this paper is to introduce the notion of n-ary S-prime hyperideals as a new expansion of n-ary prime hyperideals. Several properties and characterizations concerning n-ary S-prime hyperideals are presented. The stability of this new concept with respect to various hyperring-theoretic constructions are studied. Furthermore, we extend this concept to n-ary S-primary hyperideals. We obtained some specific results explaining the structure.
Multiplicative hyperrings are an important class of algebraic hyperstructures which generalize rings further to allow multiple output values for the multiplication operation. Let R be a commutative multiplicative hyperring. The 2-prime hyperideals are a generalization of prime hyperideals. In this paper, we aim to study 2-prime hyperideals and give some results. Moreover, we investigate delta-2-primary hyperideals which are an expansion of 2-prime hyperideals.
Various classes of hyperideals have been studied in many papers in order to let us fully understand the structures of hyperrings in general. The purpose of this paper is the study of some hyperideals whose concept is created on the basis of the intersection of all n-ary prime hyperideals in category of Krasner (m,n)-hyperrings. In this regard we introduce notions of N-hyperideals, delta-N-hyperideals, (k,n)-absorbing delta-N-hyperideals and S-N-hyperideals. The overall framework of these structures in then explaind, providing a number of major conclusions.
In this paper, we define the concepts of r-hyperideal and n-hyperideal of the multiplicative hyperring R which are two new classes of hyperideals. Several properties of them are provided.
Among many generalizations of primary hyperideals, weakly $n$-ary primary hyperideals and $n$-ary $S$-primary hyperideals have been studied recently. Let $S$ be an $n$-ary multiplicative set of a commutative Krasner $(m,n)$-hyperring $K$ and, $ϕ$ and $δ$ be reduction and expansion functions of hyperideals of $K$, respectively. The purpose of this paper is to introduce $n$-ary $ϕ$-$δ$-$S$-primary hyperideals which serve as an extension of $S$-primary hyperideals with the help of $ϕ$ and $δ$. We present some main results and examples explaining the sructure of this concept. We examine the relati…
Let R be a multiplicative hyperring with identity. In this paper, we define the concept of J-prime hyperideals which is a generalization of n-hyperideals and we will show some properties of them. Then we extend the notion of J-prime to quasi J-prime and 2-absorbing J-prime hyperideals. Various characterizations of them are provided.
Let R be a multiplicative hyperring. In this paper, we define the concept of 1-absorbing prime hyperideals which is a generalization of the prime hyperideals. Several properties of the hyperideals are provided. Moreover, we introduce the notions of 1-absorbing primary hyperideals, strongly 1-absorbing primary hyperideals and weakly 1-absorbing primary hyperideals which are generalizations of the 1-absorbing prime hyperideals and then we will show some properties of them.
