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 Rⁿ with m, n finite, then m = n.

(Rⁿ above also has a product ring structure in addition to that of R-module, but the isomorphism required for IBN is not required to be a ring isomorphism.)

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 Rⁿ 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 Rⁿ. 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 free 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

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.

An example of a ring that does not satisfy IBN is the ring of column finite matrices $\mathbb {CFM} _{\mathbb {N} }(R)$, the matrices with coefficients in a ring R, with entries indexed by $\mathbb {N} \times \mathbb {N}$ and with each column having only finitely many non-zero entries. That last requirement allows to define the product of infinite matrices MN, giving the ring structure. A left module isomorphism $\mathbb {CFM} _{\mathbb {N} }(R)\cong \mathbb {CFM} _{\mathbb {N} }(R)^{2}$ is given by:

${\begin{array}{rcl}\psi :\mathbb {CFM} _{\mathbb {N} }(R)&\to &\mathbb {CFM} _{\mathbb {N} }(R)^{2}\\M&\mapsto &({\text{odd columns of }}M,{\text{ even columns of }}M)\end{array}}$

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 (Hungerford).

From this isomorphism, it is possible to show (abbreviating $\mathbb {CFM} _{\mathbb {N} }(R)=S$) that S≅Sⁿ for any positive integer n, and hence Sⁿ≅S^m 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 (Abrams 2002).

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.

References

Abrams, Gene; Ánh, P. N. (2002), "Some ultramatricial algebras which arise as intersections of Leavitt algebras", J. Algebra Appl., 1 (4): 357–363, doi:10.1142/S0219498802000227, ISSN 0219-4988, MR 1950131

Hungerford, Thomas W. (1980), Algebra, Graduate Texts in Mathematics, vol. 73, New York: Springer-Verlag, pp. xxiii+502, ISBN 0-387-90518-9, MR 0600654 {{citation}}: Unknown parameter |note= ignored (help)