Abelian groups with free subgroups of infinite index and their endomorphism groups
The author considers the class \b{S} of torsion-free abelian groups A such that A is not free but every subgroup of A of infinite index is free. It is shown that every \b{S}-group A is strongly indecomposable of finite rank and contains a free subgroup F such that \(A/F\simeq Z(p^{\infty})\). Using \textit{D. Arnold}'s duality theory for quotient divisible groups [Pac. J. Math. 42, 11-15 (1972; Zbl 0262.20062)], the author establishes a one-to-one correspondence between the quasi- isomorphism classes of \b{S}-groups and so-called p-adic invariants: these are pairs (p,[B]) where p is a prime, B is a finite-rank pure subgroup of the p-adic integers \(Q^*_ p\) containing 1, and [B] is the set of all \(\alpha\) B where \(\alpha\) is a p-adic unit with \(1\in \alpha B\). Let E(A) denote the endomorphism ring of A, and suppose \(A\in \underline S\) has p-adic invariant (p,[B]). The following are shown: (1) \(Q\otimes E(A)\simeq Q\otimes E(B)\), and E(A) is a full subring of an algebraic number field whose degree over Q divides the rank of A. (2) If \([B]=\{B\}\) then E(A) and A are quasi-isomorphic. (3) If [B]\(\neq \{B\}\) then E(A) is free as an abelian group.
scientific article; zbMATH DE number 3910636