Quotient divisible mixed groups (Q2738749)

This paper continues the author's study of modules over the pseudo-rational numbers as an aid to understanding both quotient divisible groups and torsion-free Abelian groups of finite rank. We begin the review by reviewing the relevant definitions.NEWLINENEWLINENEWLINEThe pure subring of \(\prod_p\widehat\mathbb{Z}_p\) generated by 1 and \(\bigoplus_p\widehat\mathbb{Z}_p\) is called the ring of pseudo-rational numbers. An Abelian group \(A\) is called quotient divisible if it does not contain torsion divisible subgroups, but does contain a free finite-rank subgroup \(F\) such that \(A/F\) is torsion divisible.NEWLINENEWLINENEWLINEThree categories play important roles in the paper: 1. The category \(\mathcal{QD}\) has objects the quotient divisible groups and morphisms are the quasi-homomorphisms, \(\mathbb{Q}\otimes\text{Hom}(A,B)\). 2. The category \(\mathcal{QTF}\) has objects the torsion-free Abelian groups of finite rank and morphisms are again the quasi-homomorphisms. 2. The category \(\mathcal{FR}\) has objects that are pairs \((F,M)\), where \(F\) is a free finite-rank Abelian group contained in an \(R\)-module \(M\) such that \(F\) generates \(M\) as an \(R\)-module. A morphism from \((F,M)\) to \((F',M')\) is a pair \((f,\phi)\) such that \(f\colon F\to F'\) is a group quasi-homomorphism and \(\phi\colon M/F\to M'/F'\) is an \(R\)-module quasi-homomorphism such that \(\pi'f=\phi\pi\), where \(\pi\), \(\pi'\) denote the factor maps \(M\to M/F\) and \(M'\to M'/F'\).NEWLINENEWLINENEWLINEIt turns out that all quotient divisible groups are determined by the objects in \(\mathcal{FR}\).NEWLINENEWLINENEWLINETheorem 2.8. Let \(A\) be a quotient divisible group with a free subgroup \(F\) such that \(A/F\) is torsion-divisible. Then there exists an \(R\)-module \(M\) such that \(A\subset M\), \(F\) generates \(M\) as an \(R\)-module, and \(A\) is the pure hull of \(F\) in \(M\).NEWLINENEWLINENEWLINEThe next theorem formalizes the relationship between quotient divisible groups and objects of \(\mathcal{FR}\).NEWLINENEWLINENEWLINETheorem 3.5. The categories \(\mathcal{FR}\) and \(\mathcal{QD}\) are equivalent.NEWLINENEWLINENEWLINEHaving shown in an earlier paper with W. J. Wickless that \(\mathcal{QD}\) and \(\mathcal{QTF}\) are dual, the author immediately concludes:NEWLINENEWLINENEWLINETheorem 4.1. The category \(\mathcal{QTF}\) is dual to the category \(\mathcal{FR}\).NEWLINENEWLINENEWLINETwo more definitions introduce the final theorem. Using the duality of Theorem 4.1, the pseudo-rational rank of a torsion-free Abelian group \(A\) of finite rank is defined in terms of a rank function (not defined here) on objects in the category \(\mathcal{FR}\). The group \(A\) is called a minimax group if it contains a subgroup that satisfies the maximum condition on subgroups for which the factor group satisfies the minimum condition.NEWLINENEWLINENEWLINETheorem 4.5. The following statements are equivalent for a torsion-free finite-rank group \(A\): 1. The pseudo-rational rank of \(A\) is \(0\), 2. \(A\) is an extension of a free group by a torsion divisible group of finite rank, 3. \(A\) is a minimax group.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].