The Classification of $\mathbb{Z}_p$-Modules with Partial Decomposition Bases in $L_{\infty\omega}$

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