TOPICS
Search

Free Module


The free module of rank n over a nonzero unit ring R, usually denoted R^n, is the set of all sequences {a_1,a_2,...,a_n} that can be formed by picking n (not necessarily distinct) elements a_1, a_2, ..., a_n in R. The set R^n 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,

 (a_1,a_2,...,a_n)+(b_1,b_2,...,b_n)=(a_1+b_1,a_2+b_2,...,a_n+b_n),
(1)

2. One can multiply any sequence with any element of R according to the rule

 a(a_1,a_2,...,a_n)=(aa_1,aa_2,...,aa_n),
(2)

and this product fulfils both the associative and the distributive law.

The term free module extends to all modules which are isomorphic to R^n, i.e., which have essentially the same structure as R^n. Note that not all modules are free. For example, the quotient ring Z_m=Z/mZ, where m is an integer greater than 1 is not free, since it is a Z-module having m elements, and therefore it cannot be isomorphic to any of the modules Z^n, which are all infinite sets. Hence it is not free as a Z-module, while, of course, it is free as a module over itself.

A free module of rank n can be constructed over the ring R from any abstract set T={t_1,t_2,...,t_n} by simply taking all formal linear combinations of the elements of T with coefficients in R

 a_1t_1+a_2t_2+...+a_nt_n,
(3)

and defining the following addition

 (a_1t_1+a_2t_2+...+a_nt_n)+(b_1t_1+b_2t_2+...+b_nt_n) 
 =(a_1+b_1)t_1+(a_2+b_2)t_2+...+(a_n+b_n)t_n,
(4)

and the multiplication

 a(a_1t_1+a_2t_2+...+a_nt_n)=(aa_1)t_1+(aa_2)t_2+...+(aa_n)t_n.
(5)

The module thus obtained is often denoted by R<T>. It is generated by t_1,t_2,...,t_n, which are independent objects: this explains why it deserves to be called free. In the particular case where R is a field, R<T> is an abstract vector space having the set T 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 M generated by its subset U={u_1,...,u_n}, the map defined by a_1t_1+a_2t_2+...+a_nt_n|->a_1u_1+a_2u_2+...+a_nu_n is evidently a surjective module homomorphism from R<T> to M. This property can be generalized to all modules M, since it is easy to make it work even if the generating set U of M is infinite: it suffices to take a set T equipotent to U, and to define R<T> as the free module of "infinite rank" formed by all linear combinations

 sum_(t in T)a_tt,
(6)

in which all but finitely many of the coefficients a_t are equal to zero. The module R<T> is then isomorphic to the module direct sum

  direct sum _TR.
(7)

Note that if T is a finite set with n elements, this module is precisely R^n.


See also

Abelian Group, Abstract Vector Space, Basis, Cofree Module, Free, Free Product, Module, Ring

This entry contributed by Margherita Barile

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Free Module

Cite this as:

Barile, Margherita. "Free Module." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FreeModule.html

Subject classifications