Bases of a Module

Like a vector spaceMathworldPlanetmath over a field, one can define a basis of a module M over a general ring R with 1. To simplify matter, suppose R is commutativePlanetmathPlanetmathPlanetmath with 1 and M is unital. A basis of M is a subset B={biiI} of M, where I is some ordered index setMathworldPlanetmathPlanetmath, such that every element mM can be uniquely written as a linear combinationMathworldPlanetmath of elements from B:


such that all but a finite number of ri=0.

As the above example shows, the commutativity of R is not required, and M can be assumed either as a left or right module of R (in the example above, we could take M to be the left R-module).

However, unlike a vector space, a module may not have a basis. If it does, it is a called a free moduleMathworldPlanetmathPlanetmath. Vector spaces are examples of free modules over fields or division rings. If a free module M (over R) has a finite basis with cardinality n, we often write Rn as an isomorphicPlanetmathPlanetmathPlanetmath copy of M.

Suppose that we are given a free module M over R, and two bases B1B2 for M, is


We know that this is true if R is a field or even a division ring. But in general, the equality fails. Nevertheless, it is a fact that if B1 is finite, so is B2. So the finiteness of basis in a free module M over R is preserved when we go from one basis to another. When M has a finite basis, we say that M has finite rank (without saying what rank is!).

Now, even if M has finite rank, the cardinality of one basis may still be different from the cardinality of another. In other words, Rm may be isomorphic to Rn without m and n being equal.

Invariant Basis Number

A ring R is said to have IBN, or invariant basis number if whenever RmRn where m,n<, m=n. The positive integer n in this case is called the rank of module M. To rephrase, when F is a free R-module of finite rank, then R has IBN iff F has unique finite rank. Also, R has IBN iff all finite dimensional invertible matrices over R are square matricesMathworldPlanetmath.


  1. 1.

    If R is commutative, then R has IBN.

  2. 2.

    If R is a division ring, then R has IBN.

  3. 3.

    An example of a ring R not having IBN can be found as follows: let V be a countably infiniteMathworldPlanetmath dimensional vector space over a field. Let R be the endomorphism ringMathworldPlanetmath over V. Then R=RR and thus Rm=Rn for any pairs of (m,n).

Title IBN
Canonical name IBN
Date of creation 2013-03-22 14:51:45
Last modified on 2013-03-22 14:51:45
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 16P99
Synonym invariant basis number
Synonym invariant dimension property
Related topic ExampleOfFreeModuleWithBasesOfDiffrentCardinality
Defines basis of a module
Defines finite rank
Defines rank of a module