Invariant basis number

Invariant basis number

In mathematics, the invariant basis number (IBN) property of a ring "R" is the property that all free modules over "R" are similarly well-behaved as vector spaces, with respect to the uniqueness of their ranks.

Definition

A ring "R" has invariant basis number (IBN) if whenever the free left "R"-module "R""m" is isomorphic to "R""n" with "m", "n" finite, then "m" = "n".

Discussion

The main purpose of the invariant 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""n" over an IBN ring "R" is defined to be the cardinality of the exponent "m" of any (and therefore every) "R"-module "R""m" isomorphic to "R""n". 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 dimension. Thus the result above is in short: the rank is uniquely defined for all "R"-modules iff it is uniquely defined for finitely generated free "R"-modules.

Although in the definition above "R""m" is viewed as a "left" "R"-module, if a ring has invariant basis number with respect to left "R"-modules, it also has IBN with respect to right "R"-modules.

Examples

Clearly any field satisfies IBN. Moreover, any commutative ring satisfies IBN, as does any left-Noetherian ring and any group ring. In fact, most rings one encounters satisfy IBN.

An example of a ring that does not satisfy IBN is "R" = End"K"("V"), where "V" is an infinite dimensional vector space over field "K". To see this, write an isomorphism "V" + "V" to "V" (thus the dimension of "V" should be Dedekind infinite), which leads to an "R"-linear isomorphism "R"4 = End"K"("V+V") to "R".

Other results

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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