This paper is concerned with
the ring structures supported by certain mixed abelian groups. A class ℳ of
mixed abelian groups of torsion-free rank one is introduced, and properties of
rings on groups in ℳ are discussed. We provide complete descriptions of the
absolute annihilator and the absolute radical of groups in ℳ. These absolute
ideals are also investigated for cotorsion groups and reduced algebraically
compact groups, thus providing a partial solution to Problem 94 of Fuchs
(Infinite abelian groups, Vol. II). The results also allow us to answer a question
raised by Rotman (J. Algebra, 9 (1968), 369–387) concerning completions of
rings.
