Commutative group algebras of thick Abelian \(p\)-groups. (Q2581098)

Let \(RG\) be the group ring of an Abelian \(p\)-group \(G\) over a commutative ring \(R\) with identity and let \(S(RG)\) be the \(p\)-component of the group of the normalized units of \(RG\). The author tries to give necessary and sufficient conditions for the group \(S(RG)\) to be thick [\textit{K. Benabdallah, R. Wilson}, Can. J. Math. 30, 650-654 (1978; Zbl 0399.20049)] when the characteristic of \(R\) is \(p\) (Theorem 1) and also for \(S(KG)\) to be thick when \(K\) is a field of the first kind with respect to \(p\) of characteristic different from \(p\) (Theorem 2). However he does not succeed to prove it. The proofs of the main results are incorrect, unclear and groundless. This paper is unsuccessful since in the proofs of the above two basic theorems are used proofs of preliminary assertions which have the following essential mistakes and lapses. 1) We can give the following counterexample for the inclusion \[ G^{p^s}[p^{2n}]\setminus G^{p^s}[p^{2(n-1)}]\subseteq G^{p^n},\;n>s\geq 1, \] in the proof of Proposition 1 (page 322, line 5 from above). Let \(G=\coprod_{i=1}^\infty\langle a_i\rangle\) where the order of \(a_i\) is \(p^i\), \(i=1,2,\dots\). Then \(a_{2n+s}^{p^s}\in G^{p^s}[p^{2n}]\setminus G^{p^s}[p^{2(n-1)}]\), but \(a_{2n+s}^{p^s}\notin G^{p^n}\). Therefore, the indicated inclusion is not true. Since the incorrectly proved Proposition 1 is used for the proof of Theorem 1, then the last one remains certainly unproved. 2) The proof of the Lemma (page 325, line 4 from above) is absolutely groundless. Namely, it is not hard to see that the equality (i) ``\(g^pe=e\)'' (page 325, line 8 from above) is fulfilled if and only if (ii) \(g^p\) belongs to a finite subgroup \(A_1\) of the subgroup \(A\) of \(G\). However, \(g\) is an arbitrary element of \(G^{p^n}\setminus G[p]\) and the requirement (ii), i.e. (i), is groundless since \(G\) is an arbitrary Abelian \(p\)-group. Since this incorrectly proved Lemma is used for the proof of Theorem 2, then the last one also remains unproved. 3) In the proof of the Group Proposition the author absolutely groundlessly asserts that \(CG^1\) is a nice subgroup of \(G\) (page 321, line 2 from above). 4) In the proof of Theorem 1 the author incorrectly cites his Proposition from the year 2001 (page 323, line 2 from below) [\textit{P. V. Danchev}, C. R. Acad. Bulg. Sci. 54, No. 2, 5-8 (2001; Zbl 0972.16018), Theorem 1] although he has to cite only a result of the reviewer [\textit{T. Z. Mollov}, Publ. Math., Debrecen 19(1972), 87-96 (1973; Zbl 0273.20004), Proposition 1]. 5) In the Proposition (page 324, line 3 from below) the author incorrectly cites his paper [8] from the year 2003 [\textit{P. V. Danchev}, Serdica Math. J. 29, No. 1, 33-44 (2003; Zbl 1035.16025)] although he has to cite only a result of the reviewer [\textit{T. Z. Mollov}, PLISKA, Stud. Math. Bulg. 8, 34-46 (1986; Zbl 0662.16008), Proposition 11]. 6) The author writes ``for some ordinal \(\alpha_k\)'' (page 321, line 7 from below) although he has to write ``for some cardinal \(\alpha_k\)''. In this way the main results of the paper remain unproved.