The preceding theorem ensures that this "n" is well defined. If no such finite set of submodules exists, then u.dim("M") is defined to be ∞. When speaking of the uniform dimension of a ring, it is necessary to specify whether u.dim("R""R") or rather u.dim("R""R") is being measured. It is possible to have two different uniform dimensions on the opposite sides of a ring. If "N" is a submodule of "M", then u.dim("N") ≤  u.dim("M") with equality exactly when "N" is an essential submodule of "M". In particular, "M" and its injective hull "E"("M") always have the same uniform dimension. 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. Thus if "M" is either Noetherian or Artinian, "M" has finite uniform dimension. If "M" has finite composition length "k", then u.dim("M") ≤  k with equality exactly when "M" is a semisimple module. A standard result is that a right Noetherian domain is a right Ore domain. In fact, we can recover this result from another theorem attributed to Goldie, which states that the following three conditions are equivalent for a domain "D":  The dual notion of a uniform module is that of a hollow module: a module "M" is said to be hollow if, when "N"1 and "N"2 are submodules of "M" such that formula_6, then either "N"1 = "M" or "N"2 = "M". Equivalently, one could also say that every proper submodule of "M" is a superfluous submodule. These modules also admit an analogue of uniform dimension, called co-uniform dimension, corank, hollow dimension or dual Goldie dimension. Studies of hollow modules and co-uniform dimension were conducted in , , , and . 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.