Modules with strange decomposition properties

It is well-known that modules with pathological properties can be obtained by constructing modules with \nearly prescribed" endomorphism rings. (See, for example, [5], [2], [3], or [10].) Here we shall investigate that method using a rather weak notion of nearly prescribing the endomorphism ring, namely one that just requires that a certain ring be algebraically closed in the endomorphism ring of the module. As we shall see, in certain cases this method will su ce to construct pathological modules where it seems di cult, if not impossible, for set-theoretic reasons, to prescribe the endomorphism ring more precisely. We will focus on modulesM which are pathological in the sense that for some r 2, M =M if and only if m n (mod r). Any class, C, of modules (closed under nite direct sums and direct summands) which contains such a pathological module M for r = 2 lacks a satisfactory classi cation theorem in the sense that there are negative answers to the following two Kaplansky test problems (cf. [22, pp. 12f]):