Quasi-closed primary components in abelian group rings

Abstract

All groups considered in this work are assumed to be multiplicatively written abelian groups and all rings are commutative with identity of prime characteristic p for some fixed prime p. We follow essentially throughout the notation and terminology to the abelian group theory of the excellent classical monographs of L. Fuchs [8]. All topological references are to the p-adic topology. For G a group and R a ring, RG will denote a group ring with a normed Sylow p-subgroup S(RG), which is in the focus of our interest. The notations and terminology from the commutative group algebras theory of the nice book of G. Karpilovsky [10] will be followed. This paper is a supplement and a generalization to our previous articles [6, 7]. The main purpose that motivates the present research is the global investigation of the quasicompleteness of S(RG) for large R and G on their minimal restrictions. In particular, as corollaries to our main results, we will obtain well-documented facts in [7] and other given by us in [6]. Before proving the central theorems, we need in the sequel some assertions stated in the following paragraph.