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Like a vector space over a field, one can define a basis of a module  over a general ring  with 1. To simplify matter, suppose  is commutative with  and  is unital. A basis of  is a subset
 of  , where  is some ordered index set, such that every element  can be uniquely written as a linear combination of elements from  :
such that all but a finite number of  .
As the above example shows, the commutativity of  is not required, and  can be assumed either as a left or right module of  (in the example above, we could take  to be the left  -module).
However, unlike a vector space, a module may not have a basis. If it does, it is a called a free module. Vector spaces are examples of free modules over fields or division rings. If a free module  (over  ) has a finite basis with cardinality  , we often write  as an isomorphic copy of  .
Suppose that we are given a free module  over  , and two bases
 for  , is
We know that this is true if  is a field or even a division ring. But in general, the equality fails. Nevertheless, it is a fact that if  is finite, so is  . So the finiteness of basis in a free module  over  is preserved when we go from one basis to another. When
 has a finite basis, we say that  has finite rank (without saying what rank is!).
Now, even if  has finite rank, the cardinality of one basis may still be different from the cardinality of another. In other words,  may be isomorphic to  without  and  being equal.
A ring  is said to have IBN, or invariant basis number if whenever
 where
 ,  . The positive integer  in this case is called the rank of module  . To rephrase, when  is a free  -module of finite rank,
then  has IBN iff  has unique finite rank. Also,  has IBN iff all finite dimensional invertible matrices over  are square matrices.
Examples
- If
 is commutative, then  has IBN.
- If
 is a division ring, then  has IBN.
- An example of a ring
 not having IBN can be found as follows: let  be a countably infinite dimensional vector space over a field. Let  be the endomorphism ring over  . Then
 and thus  for any pairs of  .
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"IBN" is owned by CWoo.
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(view preamble)
Cross-references: endomorphism ring, countably infinite, square matrices, matrices, invertible, finite dimensional, iff, integer, positive, rank, equality, even, bases, isomorphic, cardinality, division rings, examples of free module, free module, module, right module, number, finite, linear combination, index set, subset, basis, unital, commutative, ring, field, vector space
There are 8 references to this entry.
This is version 9 of IBN, born on 2004-11-29, modified 2008-06-05.
Object id is 6537, canonical name is IBN.
Accessed 4266 times total.
Classification:
| AMS MSC: | 16P99 (Associative rings and algebras :: Chain conditions, growth conditions, and other forms of finiteness :: Miscellaneous) |
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Pending Errata and Addenda
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