Hereditary rings integral over their centers
type: Item
instance of: scholarly article
title: Hereditary rings integral over their centers
zbMATH Open document ID: 0587.16003
DOI: 10.1016/0021-8693(86)90130-4
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Throughout this review R will denote a right hereditary ring with centre C. It was shown by Bergman that C need not be hereditary, and indeed C need not be hereditary even when R is also a right Noetherian P.I. ring [\textit{L. W. Small} and \textit{A. R. Wadsworth}, Commun. Algebra 9, 1105- 1118 (1981; Zbl 0453.16008)]. On the other hand certain extra conditions on R do force C to be hereditary, e.g. when R is a finite C-module [\textit{S. Jøndrup}, J. Lond. Math. Soc., II. Ser. 15, 211-212 (1977; Zbl 0353.16012)]. In the present paper it is shown that if R is integral over C then C is hereditary. It is also shown that if (1) R is integral over C, (2) the non-zero elements of C are not zero-divisors in R, and (3) C is not a field, then R is a direct sum of left and right Noetherian hereditary prime rings. In particular, if R is prime right hereditary integral over its centre C and C is not a field then R is left and right Noetherian, but if C is a field then an example shows that R need not be Noetherian.
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16U30
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 3941725
zbMATH Keywords: right hereditary ring
zbMATH Keywords: centre
zbMATH Keywords: direct sum of left and right Noetherian hereditary prime rings
zbMATH Keywords: prime right hereditary integral over its centre
author: Ellen Kirkman
author: James Kuzmanovich
reviewed by: A. W. Chatters
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90130-4
OpenAlex ID: W2041477515
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description: scientific article; zbMATH DE number 3941725