Carriers of torsion-free groups
The carrier of a torsion-free abelian group A is defined to be the set \(c(A)=\{p\in \Pi |\) pA\(\neq A\}\) where \(\Pi\) denotes the set of all rational primes. The number-theoretical properties of the primes in c(A) had played a rôle in the authors' earlier investigations on subgroups of algebraic number fields and division algebras [Pac. J. Math. 109, 165- 177 (1983; Zbl 0533.16009), Lect. Notes Math. 1006, 49-96 (1983; Zbl 0515.12006); Houston J. Math. 17, 207-226 (1991)]. Given an algebraic number field F whose ring of algebraic integers is \({\mathcal O}_ F\), let I(F) denote the set of all \({\mathcal O}_ F\)-submodules A of F which contain 1 and have the property that their quasi-endomorphism ring consists of the left multiplications by elements in F. Those members of I(F) which are closed under multiplication constitute a set of subrings of F which is denoted by E(F). The authors investigate which subsets \(\Xi\) of \(\Pi\) are carriers of groups in I(F) and which \(\Xi\) are carriers of rings in E(F). Their main results are: (1) a set \(\Xi\) of rational primes is the carrier of a group in I(F) if and only if \(\Xi\) is the carrier of a ring in E(F); and (2) in the case that F is a normal extension of \({\mathbb{Q}}\), the structure of the Galois group of F over \({\mathbb{Q}}\) determines, up to finitely many primes, those subsets of \(\Pi\) which are carriers of groups in I(F).
scientific article; zbMATH DE number 4212327