On complemented radicals
type: Item
instance of: scholarly article
title: On complemented radicals
zbMATH Open document ID: 0911.16013
published in: Journal of Algebra
publication date: 1999-05-04T00:00:00Z
review text: The authors obtain an explicit description of complemented radicals in the lattice of all radicals. It follows from their description that a radical \(\alpha\) is complemented in the lattice of all radicals if and only if either \(\alpha\) or its complement is a special radical determined by a finite family of matrix rings over finite fields. An ideal \(I\) of a ring \(A\) is essential if it has a nonzero intersection with any nonzero ideal of \(A\). Given an abstract class \(\alpha\) and a ring \(A\) with an ideal \(I\), then \(I\) is an \(\alpha\)-ideal if \(I\in\alpha\). Set \({\mathcal E}\alpha=\{A\mid A\) has an essential \(\alpha\)-ideal\}. The class \({\mathcal E}\alpha\) is called the essential cover of \(\alpha\). Denote the semisimple class of \(\alpha\) by \({\mathcal S}\alpha\). The authors study radicals \(\alpha\) satisfying one of the following conditions: (I) If \(A\) is a subdirect product of rings belonging to \({\mathcal E}\alpha\), then \(\alpha(A)\neq 0\). (II) If \(A\) is a subdirect product of rings belonging to \({\mathcal E}\alpha\), then \(\alpha(A)\) is an essential ideal of \(A\). The authors obtain four results about such rings: 1. If a radical \(\alpha\) satisfies (I), then either \(\alpha\) or \({\mathcal S}\alpha\) satisfies some proper polynomial identity. 2. If a radical \(\alpha\) satisfies (II), then either every finitely generated \(\alpha\)-semisimple ring is a direct sum of a finite number of matrix rings over finite fields, or for every finitely generated ring \(R\) there exists a central idempotent \(e\in R\) such that \(\alpha(R)=eR\). Furthermore, \(\alpha\) is complemented in the lattice of all radicals. 3. The authors give an explicit description of radicals \(\alpha\) with semisimple \({\mathcal E}\alpha\). They show that \({\mathcal E}\alpha\) is semisimple if and only if \(\alpha\) is hereditary and satisfies (I). 4. They obtain an explicit description of radicals \(\alpha\) such that every finitely generated \(\alpha\…
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 1211666
zbMATH Keywords: essential covers
zbMATH Keywords: generalized polynomial identities
zbMATH Keywords: lattices of radicals
zbMATH Keywords: complemented radicals
zbMATH Keywords: special radicals
zbMATH Keywords: matrix rings over finite fields
zbMATH Keywords: semisimple classes
zbMATH Keywords: subdirect products
zbMATH Keywords: direct sums of simple rings
author: K. I. Beidar
author: Y. Fong
author: Wen-Fong Ke
MaRDI profile type: Publication
full work available at URL: jabr.1997.7254
OpenAlex ID: W2062011423
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DOI: 10.1006/JABR.1997.7254
description: scientific article; zbMATH DE number 1211666