Commutative group algebras of \(p^{\omega+n}\)-projective Abelian groups.
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, \(g_2^p\) belongs to the subgroup generated by \(g_i^{(1)}\) and this subgroup is contained in \((G\setminus G^p)\cup G^{p^\omega}\). This is a contradiction since \(g_2^p\in(G^p\setminus G^{p^\omega})\). (ii) There is an essential mistake in the proof of Proposition 4 since the author assumes that \(G[p^n]\), \(n\geq 2\), is a valuated vector space over the simple field \(\mathbb{F}_p\). Proposition 4 is used in the proofs of Theorems 4 and 5. (iii) In Theorem 4 the author cannot prove that \(H\) is a splitting group. He concludes without reason that if \(H_p\) is isomorphic to a direct factor of \(V(\mathbb{F}_pH)\) then \(H_p\) is a direct factor of \(V(\mathbb{F}_pH)\). The analogous defect the author makes in the proof of Theorem 7 of the paper [14] = [Commun. Algebra 28, No. 5, 2521-2531 (2000; Zbl 0958.20003)].
scientific article; zbMATH DE number 2164429