On May Modules of Finite Rank and the Jacobson Radicals of Their Endomorphism Rings

In [5], W. May studied the question of when isomorphisms of the endomorphism rings of mixed modules are necessarily induced by isomorphisms of the underlying modules. In so doing he introduced a class of mixed modules over a complete discrete valuation domain; in [4] these modules were renamed after their inventor. The class of May modules contains the class of Warfield modules. In this work, an intermediate class of finite rank modules is considered, called the Butler-May modules, that parallels the idea of a Butler torsion-free abelian group. Results of M. Flagg from [2] on the Jacobson radicals of the endomorphism rings of finitely generated Warfield modules are generalized to May modules. Finally, a negative example is given to an interesting and unresolved question from [2].