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.
Uniform dimension should not be confused with the related notion, also due to Goldie, of the reduced rank of a module.
Any commutative domain is a uniform ring, since if a and b are nonzero elements of two ideals, then the product ab is a nonzero element in the intersection of the ideals.
The following theorem makes it possible to define a dimension on modules using uniform submodules.
It is a module version of a vector space theorem: Theorem: If Ui and Vj are members of a finite collection of uniform submodules of a module M such that
is an essential submodule of M. The preceding theorem ensures that this n is well defined.
It is also true that u.dim(M) = n if and only if E(M) is a direct sum of n indecomposable injective modules.
It can be shown that u.dim(M) = ∞ if and only if M contains an infinite direct sum of nonzero submodules.
If M has finite composition length k, then u.dim(M) ≤ k with equality exactly when M is a semisimple module.
Studies of hollow modules and co-uniform dimension were conducted in (Fleury 1974), (Reiter 1981), (Takeuchi 1976), (Varadarajan 1979) and (Miyashita 1966).
The reader is cautioned that Fleury explored distinct ways of dualizing Goldie dimension.
Varadarajan, Takeuchi and Reiter's versions of hollow dimension are arguably the more natural ones.
Sarath and Varadarajan showed later, that M/J(M) being semisimple Artinian is also sufficient for M to have finite hollow dimension provided J(M) is a superfluous submodule of M.[1] This shows that the rings R with finite hollow dimension either as a left or right R-module are precisely the semilocal rings.