Absolute ideals of algebraically compact abelian groups. (English. Russian original) Zbl 1484.20092
J. Math. Sci., New York 259, No. 4, 444-462 (2021); translation from Fundam. Prikl. Mat. 22, No. 5, 91-114 (2019).
The authors research an absolute ideal of an abelian group \(G\), i.e. subgroups of \(G\) such that there is an ideal in every ring whose additive group coincides with \(G\). The ring is called an AI-ring if any ideal is an absolute ideal of the additive group of this ring. If there exists at least one AI-ring on an abelian group \(G\), then \(G\) is called an RAI-group. The following problem is formulated in the well-known monograph by L. Fuchs [Infinite abelian groups. Vol. II. New York-London: Academic Press (1973; Zbl 0257.20035]: chracterize RAI groups (Problem 93). The authors describe RAI-groups in the class of reduced algebraically compact abelian groups using invariants of these groups. An abelian group \(G\) is called afi-group if every absolute ideal of \(G\) is a fully invariant subgroup of \(G\). The description of afi-groups in the class of reduced algebraically compact abelian groups is obtained too.
Reviewer: Nikolay I. Kryuchkov (Ryazan)
Citations:
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