View source for Invariant basis number

{{No footnotes|date=November 2020}}
In the [[Mathematics|mathematical]] field of [[ring theory]], a [[Ring (mathematics)|ring]] ''R'' has the '''invariant basis number''' ('''IBN''') property if all finitely generated [[free module]]s over ''R'' have a well-defined rank. In the case of [[Field (mathematics)|fields]], the IBN property is the fact that finite-dimensional [[vector space]]s have a unique [[Dimension (vector space)|dimension]].

==Definition==
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 {{nowrap|1=''m'' = ''n''}}.

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>.

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 {{nowrap|1=''AB'' = ''I''}} and {{nowrap|1=''BA'' = ''I''}}, then {{nowrap|1=''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.<ref>{{harv|Lam|1999|p=3}}</ref>

Note that the isomorphisms in the definitions are ''not'' ring isomorphisms, they are module isomorphisms, even when one of ''n'' or ''m'' is 1.

==Properties==
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.

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.

==Examples==
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 the [[zero ring]]) satisfies IBN,<ref>{{cite web|url=https://stacks.math.columbia.edu/tag/0FJ7|title=Stacks Project, Tag 0FJ7|website=stacks.math.columbia.edu|accessdate=4 March 2023}}</ref> as does any [[left-Noetherian ring]] and any [[semilocal ring]].

{{hidden begin|toggle=left|title=Proof}}
Let ''A'' be a commutative ring and assume there exists an ''A''-module isomorphism <math>f \colon A^n \to A^p</math>. Let <math>(e_1,\dots,e_n)</math> the canonical basis of ''A''<sup>''n''</sup>, which means <math>e_i\in A^n</math> is all zeros except a one in the ''i''-th position. By [[Krull's theorem]], let ''I'' a [[Maximal ideal|maximal]] proper [[Ideal (ring theory)|ideal]] of ''A'' and <math>(i_1,\dots,i_n)\in I^n</math>. An ''A''-module morphism means
:<math>f(i_1,\dots,i_n) = \sum_{k=1}^n i_k f(e_k) \in I^p</math>
because ''I'' is an ideal. So ''f'' induces an ''A''/''I''-module morphism <math>f' \colon \left(\frac{A}{I}\right)^n\to \left(\frac{A}{I}\right)^p</math>, that can easily be proven to be an isomorphism. Since ''A''/''I'' is a field, ''f''' is an isomorphism between finite dimensional vector spaces, so {{nowrap|1=''n'' = ''p''}}.
{{hidden end}}

An example of a nonzero 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:
:<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>

This infinite matrix ring turns out to be isomorphic to the [[endomorphism]]s of a right [[free module]] over ''R'' of [[countable]] rank.<ref>{{harv|Hungerford|1980|p=190}}</ref>

From this isomorphism, it is possible to show (abbreviating <math>\mathbb{CFM}_\mathbb{N}(R)=S</math>) that {{nowrap|''S'' ≅ ''S''<sup>''n''</sup>}} for any positive integer ''n'', and hence {{nowrap|''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]].<ref>{{harv|Abrams|Ánh|2002}}</ref>

==Other results==
IBN is a necessary (but not sufficient) condition for a ring with no zero divisors to be embeddable in a [[division ring]] (compare [[field of fractions]] in the commutative case).  See also the [[Ore condition]].

Every nontrivial [[division ring]] or [[stably finite ring]] has invariant basis number.

Every ring satisfying the rank condition (i.e. having [[unbounded generating number]]) must have invariant basis number.<ref>{{harv|Lam|1999|loc=Proposition 1.22}}</ref>

==References==
{{reflist}}

==Sources==
* {{citation
   |last1=Abrams
   |first1=Gene
   |last2=Ánh
   |first2=P. N.
   |title=Some ultramatricial algebras which arise as intersections of Leavitt algebras
   |journal=J. Algebra Appl.
   |volume=1
   |year=2002
   |number=4
   |pages=357–363
   |issn=0219-4988
   |mr=1950131
   |doi=10.1142/S0219498802000227
}}

* {{citation
   |last=Hungerford
   |first=Thomas W.
   |title=Algebra
   |series=Graduate Texts in Mathematics
   |volume=73
   |publisher=Springer-Verlag
   |place=New York
   |year=1980
   |origyear=1974
   |pages=xxiii+502
   |isbn=0-387-90518-9
   |mr=600654
}} Reprint of the 1974 original

* {{citation
   |last=Lam
   |first=Tsit Yuen
   |title=Lectures on modules and rings
   |series=Graduate Texts in Mathematics
   |volume=189
   |publisher=Springer-Verlag
   |place=New York
   |year=1999
   |pages=xxiv+557
   |isbn=0-387-98428-3
   |mr=1653294
}}

{{DEFAULTSORT:Invariant Basis Number}}
[[Category:Module theory]]
[[Category:Commutative algebra]]
[[Category:Ring theory]]
[[Category:Homological algebra]]