Representing a ring as the endomorphism ring of an abelian group

Abstract

We investigate Baer's realization problem for almost free abelian groups, focusing on the extent to which rings can be represented as endomorphism rings under strong freeness conditions. Building on earlier work that relied on additional set-theoretic principles such as the diamond or strong black boxes, we develop new methods that significantly weaken these assumptions. The main result is obtained in ZFC (assuming a mild cardinal arithmetic configuration): for a strong limit singular cardinal $\mu$ with $\mu^+ < 2^\mu < 2^{\mu^+}$, and for a wide class of cotorsion-free rings, we construct $\mu^+$-free modules whose endomorphism rings are isomorphic to the given ring. This provides a substantial partial solution to a problem of G\"{o}bel and Trlifaj. The key innovation is the integration of $\kappa$-frame constructions with Shelah's Super Black Box, enabling a delicate diagonalization that eliminates nontrivial endomorphisms while preserving high degrees of freeness.