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 suffice to construct pathological modules where it seems difficult, if not impossible, for set-theoretic reasons, to prescribe the endomorphism ring more precisely. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.