Multiplication groups of quotient divisible Abelian groups

Abstract

A multiplication on an Abelian group [Formula: see text] is a homomorphism [Formula: see text]. The set [Formula: see text] of all multiplications on an Abelian group [Formula: see text] itself is an Abelian group with respect to addition; this group is called the multiplication group of [Formula: see text]. In the paper, the class [Formula: see text] of quotient divisible Abelian groups of rank 1 is considered. The group [Formula: see text] is described for [Formula: see text]. In the group [Formula: see text], multiplications defining comparable rings on [Formula: see text] are described for any group [Formula: see text]. The isomorphism problem is solved in [Formula: see text]: multiplications defining isomorphic rings on [Formula: see text] are described for any [Formula: see text].