An Invariant on Primary Abelian Groups with Applications to Their Projective Dimensions

Nunke’s problem for abelian $p$-groups of uncountable length

“…Proposition 3.1 ( [10], Corollary 1.20). Suppose G is a p ω1 -bounded C ω1group, and κ ∈ Q is a successor cardinal.…”

“…For a C ω1 -group G we now review how J G ⊆ Q f was defined in [10]. Suppose T ∈ Q f ; if T = ∅, let µ(T ) = ℵ 1 , and if T = ∅, let µ(T ) be its greatest element.…”

“…In [10] the following variation on Nunke's Problem was considered: If A and B are C ω1 -groups, describe exactly when A B ∈ F. If we let Q = R − {ℵ 1 } and Q f = R f ∩ Q, then for every C ω1 -group G an invariant J G ⊆ Q f was defined using techniques similar to the definition of L λ G in the countable case. It answered the above variation by establishing the following: Theorem 1.4.…”