A long standing problem in Module Theory is to describe those modules which are determined by their endomorphism ring, or by some distinguished subring. In general, the problem is known as the Baer-Kaplansky Problem, after the celebrated theorem that if \(R\) is a complete discrete valuation domain and \(M\) and \(N\) are torsion \(R\)-modules, then any ring isomorphism from \(\text{End}(M)\) onto \(\text{End}(N)\) induces a module isomorphism of \(M\) onto \(N\).
Many generalisations have been proved, the most relevant to this paper being that if \(M\) and \(N\) are Abelian \(p\)-groups with unbounded basic subgroups, then any ring isomorphism between the Jacobson radicals of their endomorphism rings implies that \(M\) and \(N\) are isomorphic.
In this paper, Flagg considers mixed modules over a complete discrete valuation ring \(R\). Her main results concern the class \(\mathcal D(R)\) of mixed \(R\)-modules \(M\) whose torsion submodule \(T\) has unbounded basic submodules and \(M/T\) is divisible. She shows that if \(M\in\mathcal D(R)\) and \(N\) is any \(R\)-module such that the Jacobson radicals of \(\text{End}(M)\) and \(\text{End}(N)\) are isomorphic, then \(N\in\mathcal D(R)\) and any such isomorphism induces an isomorphism of the modules.
Flagg also considers the class \(\mathcal F(R)\) of mixed \(R\)-modules \(M\) which have countable torsion-free rank and torsion submodule with unbounded basic submodule. If \(M\) and \(N\) in \(\mathcal F(R)\) have endomorphism rings with isomorphic Jacobson radicals, then they are isomorphic, but it is still open whether some isomorphism between them is induced.