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Concept
Uniform module
Summary
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.
Being a uniform module is not usually preserved by direct products or quotient modules. The direct sum of two nonzero uniform modules always contains two submodules with intersection zero, namely the two original summand modules. If N1 and N2 are proper submodules of a uniform module M and neither submodule contains the other, then fails to be uniform, as
Uniserial modules are uniform, and uniform modules are necessarily directly indecomposable. 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 and are both essential submodules of M, then n = m.
The uniform dimension of a module M, denoted u.
Official source
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