Representable Preradicals with Enough Projectives

Perfectly Bounded Classes of Abelian Groups

“…If λ ≤ ω, then there is a unique perfectly bounded class of length λ, namely the p λ -bounded groups that are direct sums of cyclics; however, this fails when λ > ω. This parallels results of Dugas, Fay and Shelah (1987) and Keef (1995) on the behavior of classes of abelian p-groups with elements of infinite height. It also simplifies, clarifies and generalizes a result of Cutler, Mader and Megibben (1989) which states that the p ω+1 -projectives cannot be characterized using filtrations.…”

“…For example, for λ = ω + 1, all of our examples have been based upon groups H such that p ω H ∼ = Z p , H/p ω H is p ω+1 -projective, but H is not p ω+1 -projective. On the other hand, in [8,Theorem 4] it was shown that if H is such a group, H will be a summand of H H, and so it is in T ω+1 . In other words, all of our examples have been in T ω+1 .…”