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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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