On the lattice of strong radicals
type: Item
instance of: scholarly article
title: On the lattice of strong radicals
zbMATH Open document ID: 0848.16014
author: Y. Fong
author: C. S. Wang
author: K. I. Beidar
published in: Journal of Algebra
publication date: 1996-06-05T00:00:00Z
review text: In this paper the term ``radical'' means both ``radical class'' and ``radical property''. A radical \(\alpha\) is left (right) strong if every left (right) \(\alpha\)-ideal of any ring \(\mathcal R\) belongs to \(\alpha({\mathcal R})\). A radical \(\alpha\) is said to be strong if it is both left and right strong. It is well-known that the class of all radicals forms a complete lattice with respect to set inclusion. Various authors, including Sands and Puczyłowski, have studied different aspects of this lattice and some other lattices of radicals. The author of this paper answers three open questions of Sands and Puczyłowski, namely: (i) Is the class of all strong radicals a sublattice of the lattice of all radicals? (ii) Is the class of all left strong radicals containing the prime radical a sublattice of the lattice of all radicals? (iii) If \(\alpha\) is a right strong radical containing \(\beta\) and \(J\) is a right ideal of an \(\alpha\)-semisimple ring \(\mathcal R\), does \(\alpha(J)=\beta(J)\) hold? The results of this paper give negative answers to all the above questions.
Mathematics Subject Classification ID: 16N80
zbMATH DE Number: 869781
zbMATH Keywords: strong radicals
zbMATH Keywords: lattices of radicals
zbMATH Keywords: prime radical
zbMATH Keywords: \(\alpha\)-semisimple rings
MaRDI profile type: Publication
full work available at URL: jabr.1996.0070
OpenAlex ID: W2046704834
DOI: 10.1006/JABR.1996.0070
description: scientific article; zbMATH DE number 869781