 Finite uniform decomposition of a Noetherian module

Every nonzero Noetherian module contains a uniform module: otherwise repeatedly split a nonuniform nonzero submodule into two disjoint nonzero submodules, retaining one and collecting the other, to build an infinite direct sum. Repeatedly adding a uniform submodule disjoint from the current finite sum must likewise stop. The final finite sum is an essential submodule, since any disjoint nonzero submodule would allow another addition.