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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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References:
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