Bases of a Module
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.
Invariant Basis Number
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.
If is commutative, then has IBN.
If is a division ring, then has IBN.
|Date of creation||2013-03-22 14:51:45|
|Last modified on||2013-03-22 14:51:45|
|Last modified by||CWoo (3771)|
|Synonym||invariant basis number|
|Synonym||invariant dimension property|
|Defines||basis of a module|
|Defines||rank of a module|