A survey of rings generated by units.
type: Item
instance of: scholarly article
title: A survey of rings generated by units.
zbMATH Open document ID: 1211.16010
author: Ashish K. Srivastava
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2010-10-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/115897
review text: The research topic surveyed arose from the result of K. G. Wolfson and D. Zelinsky from the 50's that every linear transformation of a vector space over a division ring but the two-element space is the sum of two units. Many surprising results have been found since then, such as construction of a von Neumann regular algebra in which not all elements are sums of units; and that the matrix algebra of at least \(2\times2\) matrices over any ring has unit sum number 3, that is, all elements are sums of exactly 3 units, due to \textit{M. Henriksen } [J. Algebra 31, 182-193 (1974; Zbl 0285.16009)]. The following research areas are described in detail: von Neumann regular rings generated additively by units; right self-injective rings generated additively by units; endomorphism rings of variously conditioned modules, such as (quasi) injective modules, generated additively by units; unit sum numbers of variously conditioned rings such as right self-injective rings and rings of integers. Furthermore, generalization of the problem with regular elements instead of units (which coincide in von Neumann regular rings) is also surveyed, and several open problems are propounded.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16-02
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 5799087
zbMATH Keywords: rings generated by units
zbMATH Keywords: sums of units
zbMATH Keywords: endomorphisms
zbMATH Keywords: von Neumann regular rings
zbMATH Keywords: right self-injective rings
zbMATH Keywords: injective modules
zbMATH Keywords: unit sum numbers
zbMATH Keywords: regular elements
MaRDI profile type: Publication
OpenAlex ID: W2088145593
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DOI: 10.5802/AFST.1281
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description: scientific article; zbMATH DE number 5799087