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'free module'
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| Title of object: |
free module |
| Canonical Name: |
FreeModule |
| Type: |
Definition |
| Created on: |
2002-01-05 16:51:02 |
| Modified on: |
2003-09-14 14:45:01 |
| Classification: |
msc:16D40 |
| Defines: |
free module, free abelian group, free basis, rank of a free module |
Revision comment (for changes between this and next version):
| Changes for correction #2623 ('rank of a free module/ other'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $R$ be a ring.
A {\it free module} over $R$
is a direct sum of copies of $R$.
Similarly, as an abelian group
is simply a module over $\Bbb{Z}$,
a {\it free abelian group}
is a direct sum of copies of $\Bbb{Z}$.
This is equivalent to saying
that the module has a {\it free basis},
i.e. a set of elements
with the property
that every element of the module
can be uniquely expressed
as an linear combination over $R$
of elements of the free basis.
In the case that a free module over $R$
is a sum of finitely many copies of $R$,
then the number of copies
is called the {\it rank} of the free module. |
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