Algebraic compactness of the \(p\)-components of the unit groups of commutative twisted group rings. (Q1881614)

Let \(R_tG\) be the twisted group ring of the Abelian group \(G\) over the commutative ring \(R\) with identity of prime characteristic \(p\), and let \(S(R_tG)\) be the \(p\)-Sylow subgroup of the unit group of the group ring. This paper deals with characterizing the algebraic compactness of the \(p\)-component \(S(R_tG)\). The analogous question for nontwisted group rings was considered by the same authors [in C. R. Acad. Bulg. Sci. 47, No. 7, 11-14 (1994; Zbl 0823.16023)]. In case \(G\) is a bounded Abelian \(p\)-group \(S(R_tG)\) is algebraically compact if and only if the \(p\)-Sylow subgroup of the group of units of the ring \(R\) is algebraically compact. Another characterization is by means of certain homomorphisms related to the \(p\)-Sylow subgroup \(G_p\) of the group \(G\) and certain homomorphisms related to the maximal divisible subgroup of \(G_p\).