2025/09/06 by Е. И. Компанцева, E., Kompantseva, Nguyen T. Q. T +1
Mathematics · #16B99 #20K30 #20K99 #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2509.05572
openalex publication_date 2025/09/06 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
The paper is devoted to the study of absolute ideals of groups in the class QD1, which consists of all quotient divisible abelian groups of torsion-free rank 1. A ring is called an AI-ring (respectively, an RF-ring) if it has no ideals except absolute ideals (respectively, fully invariant subgroups) of its additive group. An abelian group is called an RAI-group (respectively, an RFI-group) if there exists at least one AI-ring (respectively, FI-ring) on it. If every absolute ideal of an abelian group is a fully invariant subgroup, then this group is called an afi-group. It is shown that every group in QD1 is an RAI-group, an RFI-group, and an afi-group. Thus, Problem 93 of L. Fuchs' monograph ``Infinite Abelian Groups, Vol. II, New York-London: Academic Press, 1973'' is resolved within the class QD1. For any group in QD1, all rings on it that are AI-rings are described. Furthermore, the set of all AI-rings on G ∈ QD1 coincides with the set of all FI-rings on G. In addition, the principal absolute ideals of groups in QD1 are described.