On endomorphism rings of non-separable Abelian \(p\)-groups
type: Item
instance of: scholarly article
title: On endomorphism rings of non-separable Abelian \(p\)-groups
zbMATH Open document ID: 0687.20050
DOI: 10.1016/0021-8693(89)90274-3
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: A homomorphism \(\Theta:G\to H\) between Abelian \(p\)-groups \(G, H\) is thin (a notion due to Corner 1976) if for each natural number \(e\) there exists an \(n\) such that \((p^ nG[p^ e])\Theta \subseteq p^{\omega}H\). The author constructs Abelian \(p\)-groups \(G\) of \(p\)-length \(\omega +1\) such that \(\text{End\,}G=A\oplus E_{\Theta}G\), where the ring \(A\) and the cardinality \(\lambda =\lambda^{\aleph_ 0}\geq | A|\) are prescribed. In this setting \(E_{\Theta}G\) denotes all thin endomorphisms of \(G\) and \(\oplus\) is a ring split extension. The proof is based on a lifting technique using a similar result for Abelian separable \(p\)-groups. The author also discusses consequences under \(V=L\), however this proof seems not to work for length larger than or equal to \(\omega +\omega\) (as claimed without proof).
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 4125701
zbMATH Keywords: realization problem
zbMATH Keywords: Abelian \(p\)-groups
zbMATH Keywords: \(p\)-length
zbMATH Keywords: thin endomorphisms
zbMATH Keywords: separable \(p\)-groups
zbMATH Keywords: \(V=L\)
author: B. Goldsmith
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90274-3
OpenAlex ID: W1998345360
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description: scientific article; zbMATH DE number 4125701