Normed units in Abelian group rings (Q2777592)

Let \(G\) be an Abelian group whose torsion subgroup is \(tG\) and let \(F\) be a field. Suppose \(F[G]\) is the group algebra of \(G\) over \(F\), \(UF[G]\) is the unit group of \(F[G]\) and \(SF[G]\) is the \(p\)-component of \(UF[G]\). In the paper (Theorem 2.2) a description of \(UF[G]\), up to isomorphism, is given in the following partial cases: (i) when \(F\) is an algebraically closed field and the characteristic of \(F\) does not divide the orders of the elements of \(G\); (ii) when \(F\) is an algebraically closed field of prime characteristic \(p\) and \(G\) is \(p\)-splitting; (iii) when \(tG\) is finite and (iv) when \(G\) is splitting, \(tG\) is a coproduct of countable groups and \(F\) is a field of prime characteristic \(p\). These results extend some results of Chatzidakis and Pappas, the reviewer and Nachev.NEWLINENEWLINENEWLINESome more remarks of the reviewer: In case (f) of Theorem 2.2 the author incorrectly asserts: ``... the Ulm-Kaplansky cardinal functions [\textit{T. Zh. Mollov}, PLISKA, Stud. Math. Bulg. 1981, No. 2, 77-82 (1981; Zbl 0506.16008)] serve to clasify \(SF[G]\)''. For this classification a description of the maximal divisible subgroup of \(SF[G]\) has to be added. For the proof of case (e) of an arbitrary field \(F\) of prime characteristic \(p\), the author writes that he uses formulas (11) and (12), which is not admissible, since they are valid for an algebraically closed field of prime characteristic \(p\). Theorem 2.2 is not well formulated. It is overloaded with many cases and formulas. Some of them should be excluded. So, case (e) is contained in case (f) and should be skipped. Therefore, case (e) might be just a corollary of the theorem. Case (d) is auxiliary (subsidiary) and should not be formulated in Theorem 2.2. It can be a separate proposition. Formula (8) of case (a) is enough for the description of \(UF[G]\) and formulas (9) and (10) are unnecessary. Formula (12) in case (b) is also unnecessary. In case (f) the result of Chatzidakis and Pappas is used with two misprints (page 370, lines 16(+) and 16(--)). One can not agree with Claim 2.1, since in the most essential part the author only writes: ``Adapting the technique described on page 407 of [\textit{W. May}, Proc. Am. Math. Soc. 104, No. 2, 403-409 (1988; Zbl 0691.20008)], we may obtain a nice composition series for \(SF[G]\) that verifies that \(SF[G]\) is simply presented'' (page 367, lines 15(--)-13(--)). In that case nothing is explained and the main fact is not verified.