On Hopfian rings
type: Item
instance of: scholarly article
title: On Hopfian rings
zbMATH Open document ID: 0930.16003
DOI: 10.1023/A:1006655217670
author: K. Varadarajan
published in: Acta Mathematica Hungarica
publication date: 1999-10-31T00:00:00Z
review text: A ring \(R\) (associative with \(1\neq 0\)) is called Hopfian if every surjective ring homomorphism \(f\colon R\to R\) is an isomorphism. A (Hausdorff) topological space is called co-Hopfian, if every injective continuous selfmap \(f\colon X\to X\) is a homeomorphism. The main results of the paper are as follows. If \(X\) is an infinite compact totally disconnected co-Hopfian space, then the Boolean ring of clopen subsets of \(X\) is a commutative non-Noetherian Hopfian ring. If \(R\) is a Boolean Hopfian ring then the polynomial ring \(R[T]\) is Hopfian. If \(R\) and \(S\) are Hopfian rings such that \(S\) has no non-trivial central idempotents and it is not a homomorphic image of \(R\), then \(R\times S\) is Hopfian.
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 06E15
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 06E20
Mathematics Subject Classification ID: 54H10
zbMATH DE Number: 1355021
zbMATH Keywords: Hopfian rings
zbMATH Keywords: Boolean rings
zbMATH Keywords: Stone spaces
zbMATH Keywords: co-Hopfian topological spaces
zbMATH Keywords: polynomial rings
MaRDI profile type: Publication
full work available at URL: a:1006655217670
OpenAlex ID: W1835209300
description: scientific article; zbMATH DE number 1355021