 For the ring "R" = Z[√−5], both the module "R" and its submodule "M" generated by 2 and 1 + √−5 are indecomposable.While "R" is not isomorphic to "M", "R" ⊕ "R" is isomorphic to "M" ⊕ "M"; thus the images of the "M" summands give indecomposable submodules "L"1, "L"2 4 which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules, showing the analogue of the primary decomposition cannot hold for infinitely generated modules, even over the integers, Z.Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free.