Decomposition bases and Ulm’s theorem

Partial Decomposition Bases and Global Warfield Groups

“…Notice that, if G has a decomposition basis X, then w c p e G = x ∈ X U x ∈ c and U p x ∈ e (cf. [20,Corollary 3]). Lettingŵ c p e G = min w c p e G , we have the following theorem.…”

“…First, we recall the uniqueness theorem for Warfield groups. [20]). The class of Warfield groups is the unique class of groups which (1) …”

“…For a compatibility class c of Ulm matrices, prime p and equivalence class e of Ulm sequences, Stanton [20] defined the Warfield invariant of G to be w c p e G = sup rank G M /G M * p where the supremum is defined over all M ∈ c with M p ∈ e. The rank is taken to be the p-rank if e does not contain the Ulm sequence . Notice that, if G has a decomposition basis X, then w c p e G = x ∈ X U x ∈ c and U p x ∈ e (cf.…”

“…In [23], Warfield classified Warfield modules up to isomorphism by numerical invariants, the Ulm and Warfield invariants. Then Stanton [20] and Hunter and Richman [7] extended this classification theorem to global Warfield groups. Generalizing the concept of decomposition basis, the first author [8] introduced the notion of partial decomposition basis in order to extend Barwise and Eklof's [1] classification of torsion groups in L to Warfield groups.…”