Invariants of the Sylow \(p\)-subgroup of the unit group of a commutative group ring of characteristic \(p\). (English) Zbl 0828.16037

Let \(RG\) be the group ring of an Abelian group \(G\) over a commutative ring \(R\) with identity of prime characteristic \(p\) and let \(S(RG)\) be the Sylow \(p\)-subgroup of the group \(V(RG)\) of normalized units (i.e. of augmentation 1) in \(RG\). We set \(R(p)=\{r\in R\mid r^p=0\}\) and \(R[p]=\{r\in R\mid r^p=1\}\). Let \(S\) be a subring of \(R\) containing the identity of \(R\).
The following two ring-theoretic results are proved in the paper. There exist group isomorphisms: \(R(p)/S(p)\cong R[p]/S[p]\) and \(R(p)/R^p(p)\cong R[p]/R^p[p]\), where \(R(p)\) and \(R[p]\) are additive and multiplicative groups, respectively. The main result of the paper is a computation of the Ulm-Kaplansky invariants of the group \(S(RG)\). This result finishes some investigations of S. D. Berman (1967), of the reviewer (1977) and of A. A. Bovdi and Z. F. Pataj (1978).
Reviewer: T.Mollov (Plovdiv)


16U60 Units, groups of units (associative rings and algebras)
20K99 Abelian groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20E07 Subgroup theorems; subgroup growth
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[1] Berman S. D., Publ. Math. 14 pp 365– (1967)
[2] Bovdi, A. A. and Pataj, Z. F. 1978. Proc. of the Bielorussian Acad. of Sciencěs. On the construction of the centre of a multiplicative group of the group ring of thep-groups over a ring of characteristicp. 1978. Vol. 1, pp.5–11.
[3] DOI: 10.1016/0021-8693(85)90089-4 · Zbl 0597.16028 · doi:10.1016/0021-8693(85)90089-4
[4] Fuchs L., Infinite abelian groups, v.1 (1977)
[5] DOI: 10.4153/CMB-1984-031-4 · Zbl 0543.16018 · doi:10.4153/CMB-1984-031-4
[6] Mollov T. Zh, Uhm invariants of the Sylow p-subgroups of the group algebras of the abelian groups over a field of characteristic (1977) · Zbl 0506.16008
[7] Mollov T. Zh., Pliska 2 pp 77– (1981)
[8] Mollov T. Zh., Some set theory properties of the radical of Baer of commutative rings of prime characteristic (1997)
[9] Nachev N. A., Compt. Rend. Acad. Bulg. Sci.
[10] Nachev N. A., Serdica 6 pp 258– (1980)
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