In abstract algebra, 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 essential submodule. A ring may be called a right (left) uniform ring if it is uniform as a right (left) module over itself.
Alfred Goldie used the notion of uniform modules to construct a measure of dimension 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 dimension of a vector space. Finite uniform dimension was a key assumption for several theorems by Goldie, including Goldie's theorem, which characterizes which rings are right orders in a semisimple ring. Modules of finite uniform dimension generalize both Artinian modules and Noetherian modules.
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.