The classification of $${\mathbb {Z}}_p$$ Z p -modules with partial decomposition bases in $$L_{\infty \omega }$$ L ∞ ω

Abstract

Ulm's Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to L∞ω-equivalence. In this paper, we extend this classification to a class of mixed Zp-modules which includes all Warfield modules and is closed under L∞ω-equivalence. The defining property of these modules is the existence of what we call a partial decomposition basis, a generalization of the concept of decomposition basis. We prove a complete classification theorem in L∞ω using invariants deduced from the classical Ulm and Warfield invariants.

Partial Decomposition Bases and 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. Using modifications of the Ulm and Warfield invariants, abelian groups with partial decomposition bases have been completely classified in L , or equivalently, up to partial isomorphism ( [9], [10]), as well as in L ( [14], [11], [12]). …”