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| In [[mathematics]], more specifically in the field of [[ring theory]], a [[ring (mathematics)|ring]] has the '''invariant basis number (IBN)''' property if all finitely generated [[free module|free]] left [[module (mathematics)|module]]s over ''R'' have a well-defined rank. In the case of [[field (mathematics)|field]]s, the IBN property becomes the statement that finite dimensional [[vector space]]s have a unique [[dimension (vector space)|dimension]].
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| ==Definition==
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| A [[ring (mathematics)|ring]] ''R'' has '''invariant basis number''' (IBN) if for all positive integers ''m'' and ''n'', ''R''<sup>''m''</sup> [[isomorphic]] to ''R''<sup>''n''</sup> (as left ''R''-modules) implies that ''m'' = ''n''.
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| Equivalently, this means there do not exist distinct positive integers ''m'' and ''n'' such that ''R''<sup>''m''</sup> is isomorphic to ''R''<sup>''n''</sup>.
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| Rephrasing the definition of invariant basis number in terms of matrices, it says that, whenever ''A'' is an ''m'' by ''n'' matrix over ''R'' and ''B'' is an ''n'' by ''m'' matrix over ''R'' such that ''AB=1'' and ''BA=1'', then ''m=n''. This form reveals that the definition is left-right symmetric, so it makes no difference whether we define IBN in terms of left or right modules; the two definitions are equivalent.
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| Note that the isomorphisms in the definitions are ''not ring isomorphisms'', they are module isomorphisms.
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| ==Discussion==
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| The main purpose of the invariant [[basis (linear algebra)|basis]] number condition is that free modules over an IBN ring satisfy an analogue of the [[dimension theorem for vector spaces]]: any two bases for a free module over an IBN ring have the same cardinality. Assuming the [[ultrafilter lemma]] (a strictly weaker form of the [[axiom of choice]]), this result is actually equivalent to the definition given here, and can be taken as an alternative definition.
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| The '''rank''' of a free module ''R''<sup>''n''</sup> over an IBN ring ''R'' is defined to be the [[cardinality]] of the exponent ''m'' of any (and therefore every) ''R''-module ''R''<sup>''m''</sup> isomorphic to ''R''<sup>''n''</sup>. Thus the IBN property asserts that every isomorphism class of free ''R''-modules has a unique rank. The rank is not defined for rings not satisfying IBN. For vector spaces, the rank is also called the [[Hamel dimension|dimension]]. Thus the result above is in short: the rank is uniquely defined for all free ''R''-modules [[iff]] it is uniquely defined for [[finitely generated module|finitely generated]] free ''R''-modules. | |
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| ==Examples==
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| Any field satisfies IBN, and this amounts to the fact that finite dimensional vector spaces have a well defined dimension. Moreover, any [[commutative ring]] (except in the trivial case where 1 = 0) satisfies IBN, as does any [[left-Noetherian ring]] and any [[semilocal ring]].
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| An example of a ring that does not satisfy IBN is the ring of [[Matrix (mathematics)#Infinite matrices|column finite matrices]] <math>\mathbb{CFM}_\mathbb{N}(R)</math>, the matrices with coefficients in a ring ''R'', with entries indexed by <math>\mathbb{N}\times\mathbb{N}</math> and with each column having only finitely many non-zero entries. That last requirement allows us to define the product of infinite matrices ''MN'', giving the ring structure. A left module isomorphism <math>\mathbb{CFM}_\mathbb{N}(R)\cong\mathbb{CFM}_\mathbb{N}(R)^2</math> is given by:
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| :<math> \begin{array}{rcl} \psi : \mathbb{CFM}_\mathbb{N}(R) &\to & \mathbb{CFM}_\mathbb{N}(R)^2 \\ M &\mapsto & (\text{odd columns of } M, \text{ even columns of } M) \end{array}</math>
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| This infinite matrix ring turns out to be isomorphic to the endomorphisms of a right [[free module]] over ''R'' of countable rank, which is found on page 190 of {{harv|Hungerford}}.
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| From this isomorphism, it is possible to show (abbreviating <math>\mathbb{CFM}_\mathbb{N}(R)=S</math>) that ''S''≅''S''<sup>''n''</sup> for any positive integer ''n'', and hence ''S''<sup>''n''</sup>≅''S''<sup>''m''</sup> for any two positive integers ''m'' and ''n''. There are other examples of non-IBN rings without this property, among them [[Leavitt algebras]] as seen in {{harv|Abrams|2002}}.
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| ==Other results==
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| IBN is a necessary (but not sufficient) condition for a ring with no zero divisors to be embeddable in a [[division ring]] (confer [[field of fractions]] in the commutative case). See also the [[Ore condition]].
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| <references/>
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| ==References==
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| {{citation
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| |author1=Abrams, Gene
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| |author2=Ánh, P. N.
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| |title=Some ultramatricial algebras which arise as intersections of Leavitt algebras
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| |journal=J. Algebra Appl.
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| |volume=1
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| |year=2002
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| |number=4
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| |pages=357–363
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| |issn=0219-4988
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| |mr=1950131
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| |doi=10.1142/S0219498802000227
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| }}
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| {{citation
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| |author=Hungerford, Thomas W.
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| |title=Algebra
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| |series=Graduate Texts in Mathematics
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| |volume=73
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| |note=Reprint of the 1974 original
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| |publisher=Springer-Verlag
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| |place=New York
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| |year=1980
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| |pages=xxiii+502
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| |isbn=0-387-90518-9
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| |mr=600654
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| }}
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| {{DEFAULTSORT:Invariant Basis Number}}
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| [[Category:Module theory]]
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| [[Category:Commutative algebra]]
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| [[Category:Ring theory]]
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| [[Category:Homological algebra]]
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