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Topologically pure and basis subgroups in commutative group rings. (English) Zbl 0853.16040
Let $$RG$$ be the group ring of an abelian group $$G$$ over a commutative ring $$R$$ with identity of prime characteristic $$p$$ and let $$V(RG)$$ be the group of normalized units, i.e. of augmentation 1. Let $$S(RG)$$ be the $$p$$-component of $$V(RG)$$. A topologically pure subgroup in a group is introduced, as well as a subgroup $$S(RG;H)$$ of $$S(RG)$$ where $$H$$ is a subgroup of $$G$$. The paper is an announcement of the following major results. In cases of some restrictions on the subgroup $$H$$, necessary and sufficient conditions are given under which $$S(RG;H)$$ is (i) topologically pure in $$S(RG)$$, (ii) direct sum of cyclic groups, (iii) basic subgroup in $$S(RG)$$ and (iv) totally projective group. As a corollary of (iv) necessary and sufficient conditions are given under which $$S(RG)$$ is totally projective, when $$G$$ is an abelian $$p$$-group with countable limit length. The author does not mention that this assertion is connected with a result of W. May when the ring $$R$$ is perfect [Proc. Am. Math. Soc. 76, 31-34 (1979; Zbl 0406.20043)].
Some more remarks of the reviewer. The introduced subgroup $$S(RG;H)$$ has auxiliary character and it is the group $$S(RG)$$ itself that is important to investigate. The author does not mention that Corollary 1 of Section 1 for $$S(RG)$$ coincides with a result of N. A. Nachev and the reviewer [C. R. Acad. Bulg. Sci. 47, No. 7, 11-14 (1994; Zbl 0823.16023)] nor that the first assertion for $$V(RG)$$ in Theorem 2 is proved by the reviewer [Publ. Math. 18, 9-21 (1971; Zbl 0297.20016)]. Assertions (b) and (c) of Lemma 6 of Section 1, and consequently (a), are elementary and well known. Assertion (b) is used by A. Bovdi and Z. Pataj [Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 1978, No. 1, 5-11 (1978; Zbl 0394.16009)] and the first part of (c) of Lemma 6 is proved by the reviewer [Serdica 2, 219-235 (1976; Zbl 0358.20002)].
Reviewer: T.Mollov (Plovdiv)

##### MSC:
 16U60 Units, groups of units (associative rings and algebras) 20E07 Subgroup theorems; subgroup growth 16S34 Group rings 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20K10 Torsion groups, primary groups and generalized primary groups 20K25 Direct sums, direct products, etc. for abelian groups 20K40 Homological and categorical methods for abelian groups