A previously unrecognized class of a priori indecomposable modules, containing all the commonly known indecomposable modules over an Artin ring, is studied. Modules in this class are termed modules with cores. Amalgamating modules with cores, other indecomposable modules, not necessarily having cores, are constructed. With a severe restriction on the underlying ring, modules with cores are classified in terms of certain iterated amalgamations; and when every indecomposable module has a core, the underlying rings are also classified.