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| 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.
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| [[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 order]]s in a [[semisimple ring]]. Modules of finite uniform dimension generalize both [[Artinian module]]s and [[Noetherian module]]s.
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| 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.
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| ==Properties and examples of uniform modules==
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| 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 ''N''<sub>1</sub> and ''N''<sub>2</sub> are proper submodules of a uniform module ''M'' and neither submodule contains the other, then <math>M/(N_1\cap N_2)</math> fails to be uniform, as
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| :<math>N_1/(N_1\cap N_2)\cap N_2/(N_1\cap N_2)=\{0\}.</math>
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| [[Uniserial module]]s 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.
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| ==Uniform dimension of a module==
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| 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:
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| Theorem: If ''U''<sub>i</sub> and ''V''<sub>j</sub> are members of a finite collection of uniform submodules of a module ''M'' such that <math>\oplus_{i=1}^nU_i</math> and <math>\oplus_{i=1}^mV_i</math> are both [[essential submodule]]s of ''M'', then ''n'' = ''m''.
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| The '''uniform dimension''' of a module ''M'', denoted u.dim(''M''), is defined to be ''n'' if there exists a finite set of uniform submodules ''U''<sub>i</sub> such that <math>\oplus_{i=1}^nU_i</math> is an essential submodule of ''M''. 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''<sub>''R''</sub>) or rather u.dim(<sub>''R''</sub>''R'') is being measured. It is possible to have two different uniform dimensions on the opposite sides of a ring.
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| 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 module]]s.
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| 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 series|composition length]] ''k'', then u.dim(''M'') ≤ k with equality exactly when ''M'' is a [[semisimple module]]. {{harv|Lam|1999}}
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| 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'':
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| *''D'' is right Ore
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| *u.dim(''D''<sub>''D''</sub>) = 1
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| *u.dim(''D''<sub>''D''</sub>) < ∞
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| ==Hollow modules and co-uniform dimension==
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| The [[dual (mathematics)|dual]] notion of a uniform module is that of a '''hollow module''': a module ''M'' is said to be hollow if, when ''N''<sub>1</sub> and ''N''<sub>2</sub> are submodules of ''M'' such that <math>N_1+N_2=M</math>, then either ''N''<sub>1</sub> = ''M'' or ''N''<sub>2</sub> = ''M''. Equivalently, one could also say that every proper submodule of ''M'' is a [[superfluous submodule]].
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| 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 {{harv|Fleury|1974}}, {{harv|Reiter|1981}}, {{harv|Takeuchi|1976}}, {{harv|Varadarajan|1979}} and {{harv|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. Grzeszczuk and Puczylowski in {{harv|Grzeszczuk|Puczylowski|1984}} gave a definition of uniform dimension for modular lattices such that the hollow dimension of a module was the uniform dimension of its dual lattice of submodules.
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| It is always the case that a [[finitely cogenerated module]] has finite uniform dimension. This raises the question: does a [[finitely generated module]] have finite hollow dimension? The answer turns out to be no: it was shown in {{harv|Sarath|Varadarajan|1979}} that if a module ''M'' has finite hollow dimension, then ''M''/''J''(''M'') is a [[semisimple module|semisimple]], [[Artinian module]]. There are many rings with unity for which ''R''/''J''(''R'') is not semisimple Artinian, and given such a ring ''R'', ''R'' itself is finitely generated but has infinite hollow dimension.
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| 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''.<ref>The same result can be found in {{harv|Reiter|1981}} and {{harv|Hanna|Shamsuddin|1984}}</ref> This shows that the rings ''R'' with finite hollow dimension either as a left or right ''R''-module are precisely the [[semilocal ring]]s.
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| An additional corollary of Varadarajan's result is that ''R''<sub>''R''</sub> has finite hollow dimension exactly when <sub>''R''</sub>''R'' does. This contrasts the finite uniform dimension case, since it is known a ring can have finite uniform dimension on one side and infinite uniform dimension on the other.
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| ==Textbooks==
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| *{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | id={{MathSciNet|id=1653294}} | year=1999}}
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| ==Primary sources==
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| {{Reflist}}
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| *{{citation
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| |author=Fleury, Patrick
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| |title=A note on dualizing Goldie dimension
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| |journal=Canadian Mathematical Bulletin
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| |volume=17
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| |year=1974
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| |pages=511–517
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| }}
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| *{{citation
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| |author=Goldie, A. W.
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| |title=The structure of prime rings under ascending chain conditions
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| |journal=Proc. London Math. Soc. (3)
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| |volume=8
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| |year=1958
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| |pages=589–608
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| |issn=0024-6115
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| |review=MR0103206 (21 \#1988)}}
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| *{{citation
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| |author=Goldie, A. W.
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| |title=Semi-prime rings with maximum condition
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| |journal=Proc. London Math. Soc. (3)
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| |volume=10
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| |year=1960
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| |pages=201–220
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| |issn=0024-6115
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| |review=MR0111766 (22 \#2627)}}
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| *{{citation
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| |author1=Grezeszcuk, P
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| |author2=Puczylowski,E
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| |title= On Goldie and dual Goldie dimension,
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| |journal=Journal of Pure and Applied Algebra
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| |volume=31
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| |year=1984
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| |pages=47–55
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| |doi= 10.1016/0022-4049(84)90075-6}}
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| *{{citation
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| |author1=Hanna, A.
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| |author2=Shamsuddin, A.
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| |title=Duality in the category of modules. Applications,
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| |journal=Algebra Berichte 49
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| |publisher=Verlag Reinhard Fischer, Munchen
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| |year=1984}}
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| *{{citation
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| |author=Miyashita, Y.
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| |title=Quasi-projective modules, perfect modules, and a theorem for modular lattices
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| |journal=J. Fac. Sci. Hokkaido Ser. I (contd. as Hokkaido Journal of Mathematics)
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| |volume=19
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| |year=1966
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| |pages=86–110
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| |review=0213390 (35 \#4254)}}
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| *{{citation
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| |author=Reiter, E.
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| |title=A dual to the Goldie ascending chain condition on direct sums of submodules
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| |journal=Bull. Calcutta Math. Soc.
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| |volume=73
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| |year=1981
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| |pages=55–63}}
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| *{{citation
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| |author1=Sarath B.
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| |author2=Varadarajan, K.
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| |title=Dual Goldie dimension II
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| |journal=[[Communications in Algebra]]
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| |volume=7
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| |year=1979
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| |number=17
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| |pages=1885–1899}}
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| *{{citation
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| |author=Takeuchi, T.
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| |title=On cofinite-dimensional modules.
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| |journal=Hokkaido Journal of Mathematics
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| |volume=5
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| |number=1
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| |year=1976
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| |pages=1–43
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| |issn= 0385-4035
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| |review=0213390 (35 \#4254)}}
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| *{{citation
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| |author=Varadarajan, K.
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| |title=Dual Goldie dimension
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| |journal=Comm. Algebra
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| |volume=7
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| |year=1979
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| |number=6
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| |pages=565–610
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| |issn=0092-7872
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| |review=MR524269 (80d:16014)
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| |doi=10.1080/00927877908822364}}
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| [[Category:Module theory]]
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| [[Category:Ring theory]]
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