Abstrakt
We characterize Abelian groups with a minimal generating set: Let tau A denote the maximal torsion subgroup of A. An infinitely generated Abelian group A of cardinality x has a minimal generating set iff at least one of the following conditions is satisfied: 1. dim(A/pA) = dim(A/qA) = x for at least two different primes p, q. 2. dim(tau A/p tau A) = x for some prime number p. 3.
Sigma{dim(A/(pA + B)) vertical bar dim(A/(pA + B)) < x} = x for every finitely generated subgroup B of A. Moreover, if the group A is uncountable, property (3) can be simplified to (3') Sigma{dim(A/pA) vertical bar dim(A/pA) < x} = x, and if the cardinality of the group A has uncountable cofinality, then A has a minimal generating set iff any of properties (1) and (2) is satisfied.