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 $$L_{\infty \omega }$$Lźź-equivalence. In this paper, we extend this classification to a class of mixed $${\mathbb {Z}}_p$$Zp-modules which includes all Warfield modules and is closed under $$L_{\infty \omega }$$Lźź-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 $$L_{\infty \omega }$$Lźź using invariants deduced from the classical Ulm and Warfield invariants.