In [i] one has introduced the algebra QT of ~-adic numbers, depending on the type (of a torsion-free group of rank i) T and representing the generalization of the field of p-adic numbers and of Kulikov's ring of universal numbers [2]. Also there one has defined the T-adic completion A T of an Abelian torsion-free group A, of finite rank, which is a Q~-module.Since there exists a natural homomorphism A + At, it follows that by fixing in A a maximal linearly independent system x I .... , Xn, in the Q~-module Q~ one can define a submodule of
T-relationsIn the case when T coincides with the type of the group of rational numbers 6(A) coincide with the Beaumont--Pierce invariants [3] of the quotient divisible groups A and with the Murley invariants [4] for a somewhat larger class of groups.In [5] one has considered classes of groups which are determined by their modules of T-relations to within a quasiisomorphism, and one has constructed a duality which generalizes the known dualities of Warfield [6] and Arnold [7].In this paper we investigate the class ~ of torsion-free Abelian groups A of finite rank for which the module 6(A) of the T-relations is cyclic, i.e., for the groups A£~ one has in A t an equality with T-adic coefficients, ~r~+...+~I~= 0, from which all the remaining T-relations are obtained by multiplication by T-adic numbers; this entitles us to call the groups of class ~ groups with one ~-adic relation.The class ~ contains as subclasses the classes of groups, investigated previously by the author, for which any subgroup of infinite index is free [8], and the groups for which any proper servant subgroup is free [I], and also the class of groups, pointed out by A. A.Kravchenko [9] at the solution of a problem due to J. Irwin. On the other hand, the class ~, dual in the sense of [i0] to the class ~ , is a natural generalization of Murley's class of groups [4]. The intersection ~n~ coincides with the union of the class of torsionfree groups of rank 2, which benefits of a constant attention, and of the class of homogeneous, completely decomposable torsion-free groups of finite rank.In Sec. 1 the investigated classes are defined with the aid of the Richman type [ii] and, in addition, it is shown that each Abelian torsion-free group of finite rank is the sum of a finite number of groups of class ~, and also the intersection of a finite number of groups of classIn Sec. 2 we enumerate the properties of the T-adic numbers, used for the investigation of the groups of class ~,