Absolute Ideals of Murley Groups

Abstract

For an Abelian group G, a subgroup A of G is called an absolute ideal of G if A is an ideal of any ring on G. If R is a ring and any ideal of R is an absolute ideal of the additive group of R, then R is called an AI-ring. If G is an Abelian group and there exists an AI-ring on G, then G is called an RAI-group. For RAI-groups, the description problem is formulated by L. Fuchs. Obviously, every full invariant subgroup of an Abelian group G is an absolute ideal of G. E. Fried formulated the problem of studying Abelian groups for which the converse is true; i.e., every absolute ideal is a fully invariant subgroup. Such groups are called afi-groups. In this work, we describe absolute ideals of Murley groups. This allows us to describe RAI-groups, afi-groups, and E-groups in the class of Murley groups.