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Commutative group algebras of \(\sigma\)-summable Abelian groups. (English) Zbl 0886.16024
Let \(R\) be a commutative ring with identity of prime characteristic \(p\) without nilpotent elements and let \(G\) be an abelian group. We denote by \(G(p)\) the \(p\)-component of \(G\) and by \(S(RG)\) the normed \(p\)-component of the unit group of the group ring \(RG\). The author gives necessary and sufficient conditions for (i) \(S(RG)\) to be \(\sigma\)-summable and (ii) \(S(KG)\) to be \(\sigma\)-summable when \(G\) is a \(p\)-group and \(K\) is a field of the first kind with respect to \(p\). It is proved (Proposition 8) that if \(G(p)\) is a totally projective group of countable length then the isomorphism of \(RG\) and \(RH\) as \(R\)-algebras for any group \(H\) implies an isomorphism of \(H(p)\) and \(G(p)\).
Some remarks of the reviewer: The second part of the second Theorem should not be formulated as a separate result. The first assertion of the Main Lemma coincides with formulas (11) and (17), i.e. with Lemma 2.3, of the article [Commun. Algebra 23, No. 7, 2469-2489 (1995; Zbl 0828.16037)] of N. Nachev and the second assertion of this Lemma is Proposition 2 of [C. R. Acad. Bulg. Sci. 47, No. 7, 11-14 (1994; Zbl 0823.16023)]. Lemma 1 is well known and Lemma 2 is elementary. It is not noted in the paper, that the proof of Lemma 4 is analogous to the similar Lemma 4 of [Houston J. Math. 22, No. 2, 225-232 (1996; Zbl 0859.16025)]. The author does not note that the proof of the inverse part of his second Theorem exploits Proposition 11 of [PLISKA, Stud. Math. Bulg. 8, 34-46 (1986; Zbl 0662.16008)]. In the proof of the second theorem the author states that the coincidence of the maximal divisible subgroup of \(S(KG)\) with the first Ulm subgroup of \(S(KG)\) is his result although it is a result of the reviewer [Proposition 20, PLISKA, Stud. Math. Bulg. 8, 34-46 (1986; Zbl 0662.16008)].
Reviewer: T.Mollov (Plovdiv)

MSC:
16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20K10 Torsion groups, primary groups and generalized primary groups
20K99 Abelian groups
20E07 Subgroup theorems; subgroup growth
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