Uniform module

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of M is an <a href="/facts/Essential_submodule/S50wKkfs">essential submodule</a>. A ring may be called a right (left) uniform ring if it is uniform as a right (left) module over itself.
<a href="/facts/Alfred_Goldie/WBpdWTRT">Alfred Goldie</a> used the notion of uniform modules to construct a measure of <a href="/facts/Dimension/0ktmUyac">dimension</a> for modules, now known as the uniform dimension (or Goldie dimension) of a module. Uniform dimension generalizes some, but not all, aspects of the notion of the <a href="/facts/Dimension_of_a_vector_space/z8m02A3W">dimension of a vector space</a>. Finite uniform dimension was a key assumption for several theorems by Goldie, including <a href="/facts/Goldie%2527s_theorem/GBXXCQpN">Goldie's theorem</a>, which characterizes which rings are right orders in a <a href="/facts/Semisimple_ring/9KASYk35">semisimple ring</a>. Modules of finite uniform dimension generalize both <a href="/facts/Artinian_module/jbg1NSk5">Artinian modules</a> and <a href="/facts/Noetherian_module/kkxeu4Md">Noetherian modules</a>.
In the literature, uniform dimension is also referred to as simply the dimension of a module or the rank of a module. Uniform dimension should not be confused with the related notion, also due to Goldie, of the reduced rank of a module.

Uniform module open-in-new

Uniform module