The duality interchanges the subcategories of discrete groups and compact groups. If "R" is a ring and "G" is a left "R"-module, the dual group formula_3 will become a right "R"-module; in this way we can also see that discrete left "R"-modules will be Pontryagin dual to compact right "R"-modules. The ring End("G") of endomorphisms in LCA is changed by duality into its opposite ring (change the multiplication to the other order). For example, if "G" is an infinite cyclic discrete group, formula_3 is a circle group: the former has formula_131 so this is true also of the latter.