Uniform module

A nonzero module is uniform if every pair of nonzero submodules has nonzero intersection. Equivalently, every nonzero submodule is an essential submodule. A right Noetherian domain is uniform as a right module: if $aA\cap bA=0$ for nonzero $a,b$, the right ideals $b^{i}aA$, $i\ge 0$, form an infinite direct sum, contradicting the ascending chain condition.

The uniform dimension of a module is the supremum of the numbers of nonzero submodules that can occur in an internal direct sum. Finite uniform dimension is equivalent to having no infinite internal direct sum of nonzero submodules. A finite direct sum of uniform modules contained as an essential submodule gives that number as the dimension. The finite uniform decomposition of a Noetherian module therefore gives finite uniform dimension.

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.