\(\partial\)-operations and \(H_V\)-fields.
type: Item
instance of: scholarly article
title: \(\partial\)-operations and \(H_V\)-fields.
zbMATH Open document ID: 1156.16031
author: Thomas Vougiouklis
publication date: 2008-09-09T00:00:00Z
review text: The paper is devoted to the study of \(\partial\)-operations, initiated by the author [in Ratio Math. No. 15, 25-36 (2005)]. Let \((R,+,\cdot)\) be a ring and \(f,g\colon R\to R\) are mappings. The author defines the hyperoperations \((\partial_+)\) and \((\partial_\cdot)\), that are called `theta-operations', on \(R\) by \[ x\partial_+y=\{f(x)+y,x+f(y)\},\quad x\partial_\cdot y=\{g(x)y,xg(y)\} \] for all \(x,y\in R\). It is proved that \((R,\partial_+,\partial_\cdot)\) is an \(H_v\)-near-ring. Also, some other properties of this ring are investigated. In the last section of paper, examples are given from the ring of integers and it is shown that under certain conditions the resulting \(H_v\)-near-ring is an \(H_v\)-field.
reviewed by: Ali Madanshekaf
Mathematics Subject Classification ID: 16Y99
Mathematics Subject Classification ID: 20N20
Mathematics Subject Classification ID: 12K99
zbMATH DE Number: 5323446
zbMATH Keywords: hyperstructures
zbMATH Keywords: \(H_v\)-structures
zbMATH Keywords: hyperrings
zbMATH Keywords: hyperfields
zbMATH Keywords: hyper-near-rings
MaRDI profile type: Publication
full work available at URL: s10114-007-6481-5
OpenAlex ID: W1253298118
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DOI: 10.1007/S10114-007-6481-5
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5323446