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)


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