Radicals induced by the total of rings
type: Item
instance of: scholarly article
title: Radicals induced by the total of rings
zbMATH Open document ID: 0874.16014
published in: Beiträge zur Algebra und Geometrie
publication date: 1997-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/232549
full work available at URL: no.1
review text: If \(A\) is a ring, the total \(\text{Tot}(A)\) is defined as the set \(\{a\in A\mid aA\) does not contain a nonzero idempotent\}. Regretfully, \(\text{Tot}(A)\) need not be an ideal because it is not closed under addition. To overcome this difficulty, the authors consider the upper radical \(\mathcal K\) of all rings \(A\) with \(\text{Tot}(A)=0\). They call \(\mathcal K\) the Kasch radical and show that \(\mathcal K\) is a supernilpotent normal radical. The authors also study the upper radical \({\mathcal K}_p\) of all prime rings \(A\) with \(\text{Tot}(A)=0\). They show that \({\mathcal K}_p\) is a special normal radical containing the Kasch radical \(\mathcal K\). They then show that \({\mathcal K}\neq{\mathcal K}_p\). In the final part of the paper, the authors position the radicals \(\mathcal K\) and \({\mathcal K}_p\) in the lattice of all radicals. The Kasch radical is bigger than the Behrens and the Jacobson radicals, and it is incomparable with the Brown-McCoy radical, the generalized nil radical, and the strongly prime radical.
Mathematics Subject Classification ID: 16N80
zbMATH DE Number: 1018696
zbMATH Keywords: totals of rings
zbMATH Keywords: Behrens radical
zbMATH Keywords: Jacobson radical
zbMATH Keywords: lattices of radicals
zbMATH Keywords: upper radicals
zbMATH Keywords: Kasch radical
zbMATH Keywords: supernilpotent normal radicals
zbMATH Keywords: prime rings
zbMATH Keywords: special normal radicals
zbMATH Keywords: Brown-McCoy radical
zbMATH Keywords: generalized nil radical
zbMATH Keywords: strongly prime radical
author: K. I. Beidar
author: R. Wiegandt
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1018696