Free Module
The free module of rank 
over a nonzero unit ring 
, usually denoted 
, is the set of all sequences 
that can be formed by picking 
(not necessarily distinct) elements 
, 
,
..., 
in 
. The set 
is a particular example of the algebraic structure called
a module since is satisfies the following properties.
1. It is an additive Abelian group with respect to the componentwise sum of sequences,
 |
(1) |
2. One can multiply any sequence with any element of 
according to the rule
 |
(2) |
and this product fulfils both the associative and the distributive law.
The term free module extends to all modules which are isomorphic to 
, i.e., which have essentially the same structure as 
. Note that not all modules are free.
For example, the quotient ring 
, where 
is an integer greater than 1 is not free, since it is a 
-module having 
elements, and therefore it cannot be isomorphic to any of
the modules 
,
which are all infinite sets. Hence it is not free as a 
-module, while, of course, it is free as a module over itself.
A free module of rank 
can be constructed over the ring 
from any abstract set 
by simply taking all formal linear combinations
of the elements of 
with coefficients in 
 |
(3) |
and defining the following addition
 |
(4) |
and the multiplication
 |
(5) |
The module thus obtained is often denoted by 
. It is generated by 
, which are independent objects: this explains
why it deserves to be called free. In the particular case where 
is a field, 
is an abstract
vector space having the set 
as a basis.
Free modules play a central role in algebra, since any module is the homomorphic image of some free module: given a module 
generated by its subset 
, the map defined by 
is evidently
a surjective module homomorphism from 
to 
. This property can be generalized to all modules 
, since it is easy to make it work even if the generating set
of 
is infinite: it suffices to take a set 
equipotent to 
, and to define 
as the free module of "infinite rank" formed
by all linear combinations
 |
(6) |
in which all but finitely many of the coefficients 
are equal to zero. The module 
is then isomorphic to the module
direct sum
 |
(7) |
Note that if 
is a finite set with 
elements, this module is precisely 
.
See also
Abelian Group, Abstract Vector Space, Basis, Cofree Module, Free, Free Product, Module, Ring
This entry contributed by Margherita Barile
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References
Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, p. 78, 1999.Hartley, B. and Hawkes, T. O. Rings, Modules and Linear Algebra: A Further Course in Algebra Describing the Structure of Abelian Groups and Canonical Forms of Matrices Through the Study of Rings and Modules. London, England: Chapman and Hall, pp. 89-94, 1970.Kunz, E. Introduction to Commutative Algebra and Algebraic Geometry. Boston, MA: Birkhäuser, p. 14, 1985.Passman, D. S. A Course in Ring Theory. Pacific Grove, CA: Wadsworth & Brooks/Cole, pp. 16-18, 1991.Reid, M. Undergraduate Commutative Algebra. Cambridge, England: Cambridge University Press, pp. 40-41, 1995.Rowen, L. H. Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 54-56, 1988.Sharp, R. Y. Steps in Commutative Algebra, 2nd ed. Cambridge, England: Cambridge University Press, pp. 118-121, 2000.
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Cite this as:
Barile, Margherita. "Free Module." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FreeModule.html