CLASSES OF MODULES WITH MANY DIRECT-SUMMANDS

Let R be any ring with identity, M a unital right R-module and alpha greater than or equal to 0 an ordinal. Then M is a direct sum of a semisimple module and a module having Krull dimension at most a if and only if for every submodule N of M there exists a direct summand K of M such that K subset of or equal to N and N/K has Krull dimension at most alpha.