Commutative group algebras of \(p^{\omega+n}\)-projective Abelian groups.
type: Item
instance of: scholarly article
title: Commutative group algebras of \(p^{\omega+n}\)-projective Abelian groups.
zbMATH Open document ID: 1077.20005
published in: Acta Mathematica Vietnamica
publication date: 2005-05-03T00:00:00Z
review text: Let \(RG\) be the group ring of an Abelian group \(G\) over a commutative ring \(R\) of prime characteristic \(p\) and let \(V(RG)\) be the group of the normalized units of \(RG\). The paper is devoted to the direct factor problem and to the isomorphism of the group algebras of \(p^{\omega +n}\)-projective Abelian \(p\)-groups. But this article is unsuccessful since almost all of the main results have mistakes in their proofs as follows: Theorems 3, 4 and 5 and Corollaries 2 and 3. Theorem 1 has a technical value. In Theorem 2 it is proved that if \(G\) is a \(p^{\omega+1}\)-projective Abelian \(p\)-group, then \(G\) is a direct factor of \(V(RG)\) with \(p^{\omega+1}\)-projective complement. In this theorem the author proves very long (p. 264, lines 5-12 above) the obvious fact that the \(p\)-socles \(V(RG')[p]\) and \(V(RG')[p]\times G[p]\) have one and the same infinite cardinality. The other statements have the following defects: (i) For the proof of Theorem 3 we give the following counterexample. Let \(G=H_{\omega+1}\) be the Prüfer group, i.e. the Abelian group with generating elements \(a_0\) and \(a_n\) and defining relations \(a_0^p=1\), \(a_n^{p^n}=a_0\), where \(n\) is natural. We choose \(B=\langle a_0\rangle\times\langle g_1\rangle\times\langle g_2^p\rangle\), where \(g_1=a_1a_2^{-p}\) and \(g_2=a_2a_3^{-p}\). Then \(B\subseteq G[p]\). It is not hard to see that \(G/B\) is a direct sum of cyclic groups. We set \(G_1=\langle a_0\rangle\times\langle g_1\rangle\times\langle g_2\rangle\), \(G_2=G_1\) and \(G_k=\langle G_{k-1},a_k\rangle\), \(k\geq 3\). Then the groups \(G_k\) satisfy the conditions of the proof of the theorem. Since \(G^{p^\omega}=\langle a_0\rangle\), then \(g_2^p\notin G^{p^\omega}\). Suppose, as the author states, that \(G_1\subseteq M_1\). Then \(g_2^p\in G_1\subseteq M_1\). Since \(M_1\) is generated by linear combinations of the elements \(g_i^{(1)}\), then \(g_2^p\) appears to be a product of some elements \(g_i^{(1)}\). Therefore, \…
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K40
zbMATH DE Number: 2164429
zbMATH Keywords: commutative group algebras
zbMATH Keywords: isomorphism problem
zbMATH Keywords: totally projective Abelian groups
zbMATH Keywords: direct factor problem
zbMATH Keywords: groups of units
author: Peter Danchev
reviewed by: Todor Mollov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2164429