A new characterisation of hereditary PI rings
type: Item
instance of: scholarly article
title: A new characterisation of hereditary PI rings
zbMATH Open document ID: 1022.16006
DOI: 10.1007/BF02773064
author: Maria Luisa Villani
published in: Israel Journal of Mathematics
publication date: 2003-08-05T00:00:00Z
review text: The main object of study are Noetherian PI rings \(R\) with the following property (P): All proper Artinian homomorphic images of \(R\) are of finite representation type. One of the main results of the paper asserts that a right Noetherian PI ring \(R\) which is an order in an Artinian ring has the property (P) if and only if \(R\) is a product \(A\times S\) of an Artinian ring \(A\) of finite representation type and a semiprime hereditary ring \(S\). It is also shown that a Noetherian nonartinian PI ring \(R\) is a hereditary prime ring if and only if all proper homomorphic images of \(R\) are of finite representation type. Hence, it is concluded that a semiprime Noetherian PI ring \(R\) is hereditary if and only if \(R\) has the property (P). The author also proves the following generalisation of a well-known result of Warfield's. Let \(R\) be a Noetherian PI ring. Then every Artinian factor ring of \(R\) is serial if and only if \(R\) is a product \(A\times S\) of an Artinian serial ring \(A\) and a semiprime hereditary ring \(S\).
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16E60
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 1910266
zbMATH Keywords: Noetherian PI rings
zbMATH Keywords: Noetherian rings
zbMATH Keywords: semiprime rings
zbMATH Keywords: finite representation type
zbMATH Keywords: hereditary PI rings
zbMATH Keywords: orders in Artinian rings
zbMATH Keywords: serial rings
zbMATH Keywords: proper homomorphic images
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773064
OpenAlex ID: W2087447658
description: scientific article; zbMATH DE number 1910266