Definability of local abelian groups by their splitting rings (Q2211379)

The author research the problem which formulated by \textit{S. Glaz} et al. [Lect. Notes Pure Appl. Math. 171, 223--239 (1995; Zbl 0895.20046)]: When a \(p\)-local torsion-free abelian group is determined by the collection of its splitting rings? Let \(\mathcal{L}_p(K)\) be the class of \(p\)-local torsion-free abelian groups with splitting field \(K\). If \(K\) is quadratic over \(\mathbb{Q}\) then \(A\) is determined by the collection of its splitting rings if and only if the \(p\)-rank of \(A\) is equal to \(1\). The author gives necessary and sufficient conditions for definability if \(K\) is cubic over \(\mathbb{Q}\).