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单词 IBN
释义

IBN


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:

m=iIribi

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

|B1|=|B2|?

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.

Examples

  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).

TitleIBN
Canonical nameIBN
Date of creation2013-03-22 14:51:45
Last modified on2013-03-22 14:51:45
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id12
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 16P99
Synonyminvariant basis number
Synonyminvariant dimension property
Related topicExampleOfFreeModuleWithBasesOfDiffrentCardinality
Definesbasis of a module
Definesfinite rank
Definesrank of a module
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