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1	Equality
1	Division Ring
1	Matrix
1	Vector Space
1	Field
1	Positive
1	Finite
1	Set
1	Even Number
1	Integer
1	Cardinality
1	Biconditional
1	Number
1	Ring
1	Module
1	Commutative
1	Representation Lie Algebra
2	Unit
2	Fancy Definition Of Module
2	Prismatoid
2	Square Matrix
2	Basis
2	Isomorphic Groups
3	Linear Combination
3	Countably Infinite
3	Fractional Ideal Of Commutative Ring
5	Free Module
5	Index Set
7	Simple Tensor
8	Ring Of Endomorphisms
22	Example Of Free Module

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I B N

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 $B=\lbrace b_i\mid i\in I\rbrace$ of

, where

is some ordered index set, such that every element $m\in M$ can be uniquely written as a linear combination of elements from

$\displaystyle m=\sum_{i\in I}r_ib_i$

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 $B_1\neq B_2$ for , is

$\displaystyle \vert B_1\vert = \vert B_2\vert?$

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 $R^m \cong R^n$ where $m,n<\infty$ ,

. 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 $R=R\oplus R$ and thus for any pairs of .