Note on isomorphism theorems of hyperrings.
type: Item
instance of: scholarly article
title: Note on isomorphism theorems of hyperrings.
zbMATH Open document ID: 1213.16038
DOI: 10.1155/2010/376985
author: M. Velrajan
author: Arjunan Asokkumar
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2011-02-08T00:00:00Z
full work available at URL: https://eudml.org/doc/233623
review text: There are different kinds of hyperrings [\textit{B. Davvaz, V. Leoreanu-Fotea}, Hyperring theory and applications. Palm Harbor: International Academic Press (2007; Zbl 1204.16033)]. In this paper, the authors extend the isomorphism theorems of hyperrings, where the additions and multiplications are hyperoperations. In page 3, the authors give a definition for a multiplicative hyperring \((R,+,\cdot)\) which is different from the usual definition of multiplicative hyperrings. Indeed, they suppose that the strong distributive law holds in a multiplicative hyperring. But in this case, we obtain an ordinary ring instead of a hyperring. Indeed, for all \(a,b\in R\), \(|a\cdot b|=1\). Because, we have \(0=a\cdot 0\), i.e., \(|a\cdot 0|=1\). Also, we have \(0=a\cdot 0=a\cdot(b-b)=a\cdot b-a\cdot b\). Now, suppose that \(x,y\in a\cdot b\). Then, \(x-y\in a\cdot b-a\cdot b\) which implies that \(x-y\in a\cdot 0\) and so \(x=y\).
reviewed by: Bijan Davvaz
Mathematics Subject Classification ID: 16Y99
Mathematics Subject Classification ID: 20N20
zbMATH DE Number: 5847816
zbMATH Keywords: hyperoperations
zbMATH Keywords: canonical hypergroups
zbMATH Keywords: multiplicative hyperrings
zbMATH Keywords: isomorphism theorems
Wikidata QID: Q58651760
MaRDI profile type: Publication
OpenAlex ID: W2033433638
cites work: Q4065720
cites work: Q3137919
cites work: Q4706962
cites work: Q5676421
cites work: Q5604566
cites work: Q4317188
cites work: A class of hyperrings and hyperfields
cites work: Q3577618
cites work: Q3975640
cites work: Isomorphism theorems of hyperrings.
cites work: Q4870888
cites work: Q3975647
cites work: On homomorphisms of some multiplicative hyperrings.
cites work: Q4639722
cites work: Q3975656
cites work: Strongly distributive multiplicative hyperrings
cites work: A class of hyperrings
cites work: Q5384911
cites work: Q5438097
cites work: Q5193085
cites work: Q3575415
DBLP publication ID: journals/ijmmsc/VelrajanA10
description: scientific article; zbMATH DE number 5847816