Inverse-direct systems of modules have been considered by Eklof and Mekler, see [P.C. Eklof, A.H. Mekler, Almost free modules, 2nd ed., North Holland, 2002]. The systems we are going to study are different: we do not assume the condition that certain composite maps are identity maps (this forces the direct summand property). In this paper inverse-direct systems will be considered where certain composite maps lie in the center of the respective endomorphism rings. We investigate how the limits are modified if the connecting maps are changed by automorphisms of the modules. It will also be shown that one can define a composition between the systems modified by these automorphisms such that those whose limits are non-isomorphic under the canonical maps form an abelian group. This group can be described in terms of the first derived functor of the inverse limit functor. We also study the relation to vanishing inverse limits: in certain cases, the maps can be modified in such a way that the inverse limit of the new system becomes 0. In the final section, we use self-idealizations in order to construct sets of non-isomorphic modules (over suitable uncountable rings) that are direct limits of the same collection of modules with different connecting maps. © 2009.
Fuchs, L., Göbel, R., & Salce, L. (2010). On inverse-direct systems of modules. Journal of Pure and Applied Algebra, 214(4), 322–331. https://doi.org/10.1016/j.jpaa.2009.05.003