= Uniform dimension
{title2=$\operatorname{udim}M$}
The uniform dimension of a <module> is the supremum of the numbers of nonzero <submodules> that can occur in an internal <direct sum>. Finite uniform dimension is equivalent to having no infinite internal <direct sum> of nonzero <submodules>. A finite <direct sum> of <uniform modules> contained as an <essential submodule> gives that number as the dimension. The <finite uniform decomposition of a Noetherian module> therefore gives finite uniform dimension.
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