Artinian (Noetherian) part of a Goldie ring
type: Item
instance of: scholarly article
title: Artinian (Noetherian) part of a Goldie ring
zbMATH Open document ID: 0681.16009
author: K. C. Chowdhury
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1989-01-01T00:00:00Z
review text: The author studies the right (left) artinian radical (i.e. the sum of all right (left) ideals which are artinian as right (left) R-modules) of a Goldie ring R and obtains some results similar to those of the case when R is noetherian. However, if the author paid attention to the fact that a semiprime right Goldie, right quotient ring is semisimple artinian, then he would see from Lemma 2.1.2 that the statements of Theorem 3.2 and Theorem 3.4 are well-known because rings appearing in these theorems are in fact semisimple artinian. Moreover one can prove a more general result from which it follows that a right Goldie ring satisfying DCC (ACC) on essential right ideals is right artinian (right noetherian): If R is a ring satisfying DCC (ACC) on essential right ideals, then the factor ring of R by its right socle is right artinian (right noetherian). See Proposition 1.1 of \textit{E. P. Armendariz} [Commun. Algebra 8, 299-308 (1980; Zbl 0444.16015)] (resp. Lemma 2 of \textit{Nguyen V. Dung}, \textit{R. Wisbauer} and the reviewer [Arch. Math. 53, 252-255 (1989; Zbl 0658.16018)]).
reviewed by: Dinh Van Huynh
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P20
zbMATH DE Number: 4114821
zbMATH Keywords: artinian radical
zbMATH Keywords: Goldie ring
zbMATH Keywords: semiprime right Goldie
zbMATH Keywords: semisimple artinian
zbMATH Keywords: essential right ideals
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4114821