Ikeda-Nakayama rings
type: Item
instance of: scholarly article
title: Ikeda-Nakayama rings
zbMATH Open document ID: 0958.16002
published in: Journal of Algebra
publication date: 2001-01-23T00:00:00Z
review text: The starting point for this paper is a well known result of M. Ikeda and T. Nakayama that asserts that if \(R\) is a right self-injective ring and \(A\) and \(B\) are right ideals of \(R\), then \(l(A\cap B)=l(A)+l(B)\), where \(l(X)\) denotes the left annihilator of a subset \(X\) of \(R\). The authors single out this condition and call the rings satisfying it right Ikeda-Nakayama rings (right IN-rings for short). They show that if \(R\) is a right IN-ring, then for each pair \(A,B\) of right ideals of \(R\) that are complements of each other, there exists an idempotent \(e\in R\) such that \(A=eR\) and \(B=(1-e)R\). From this it follows that a right IN-ring is right quasi-continuous and that \(R\) is right self-injective if and only if the \(2\times 2\) matrix ring \(M_2(R)\) is a right IN-ring. By a result of \textit{C. R. Hajarnavis} and \textit{N. C. Norton} [J. Algebra 93, 253-266 (1985; Zbl 0595.16009)] the rings \(R\) such that every right or left ideal of \(R\) is an annihilator (called dual rings) are left and right IN-rings. Here, this result is completed by showing that \(R\) is a dual ring if and only if \(R\) is a left and right IN-ring and the dual of every simple right \(R\)-module is simple. Moreover, it is also shown that every left perfect left and right IN-ring is QF but a left and right Artinian right IN-ring need not be QF.
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16L30
zbMATH DE Number: 1448623
zbMATH Keywords: Ikeda-Nakayama rings
zbMATH Keywords: self-injective rings
zbMATH Keywords: quasi-continuous rings
zbMATH Keywords: perfect rings
zbMATH Keywords: quasi-Frobenius rings
zbMATH Keywords: annihilators
zbMATH Keywords: right IN-rings
zbMATH Keywords: idempotents
zbMATH Keywords: simple modules
author: Victor Camillo
author: W. Keith Nicholson
author: Mohamed F. Yousif
reviewed by: José L. Gómez-Pardo
MaRDI profile type: Publication
full work available at URL: jabr.1999.8217
OpenAlex ID: W2048022297
cites work: Rings satisfying certain chain conditions.
cites work: Semiperfect min-CS rings
cites work: Q4104083
cites work: On dual rings and their modules
cites work: On Some Characteristic Properties of Quasi-Frobenius and Regular Rings
cites work: Dual rings
cites work: Q3674835
cites work: Orders in self-injective semi-perfect rings
cites work: Q3995216
cites work: Principally injective rings
cites work: Mininjective rings
cites work: A generalization of quasi-Frobenius rings
cites work: On Continuous Rings and Self Injective Rings
cites work: On continuous rings
cites work: CS rings and nakayama permutations
DOI: 10.1006/JABR.1999.8217
Recommended article: Q5316086
Recommended article: The Armendariz module and its application to the Ikeda-Nakayama module.
Recommended article: Quasi-dual rings
Recommended article: Equivalence and perspectivity in Baer rings
Recommended article: Q4526386
Recommended article: On skew polynomials over Ikeda-Nakayama rings
Recommended article: Q3633666
Recommended article: On continuous rings
Recommended article: Simple rings with injectivity conditions on one-sided ideals
Recommended article: Q4009829
description: scientific article; zbMATH DE number 1448623