A free module over a nonzero unit ring is a left module with a basis, so that every element has a unique expression as a finite linear combination of basis elements with coefficients in . The standard example with a basis of elements is , the set of all sequences with entries in . It has the following operations.

1. It is an additive Abelian group with respect to the componentwise sum of sequences,

2. Any sequence can be multiplied by any element of according to the rule

and this product satisfies the associative and distributive laws.

The modules isomorphic to for some nonnegative integer are exactly the free modules with a finite basis. For example, the quotient ring , where is an integer greater than 1, is not free as a -module. It has elements, whereas a nonzero free -module is infinite and the zero free -module has only one element. It is free as a module over itself.

A free module with a basis of elements can be constructed over the ring from any set by taking all formal linear combinations of the elements of with coefficients in

and defining the following addition

and the multiplication

The module thus obtained is often denoted by . It is generated by , which are independent over . When is a field, is an abstract vector space having the set as a vector basis.

Every module is the image of some free module under a module homomorphism. Given a module generated by its subset , the map defined by is 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 subset of is infinite. Take a set with a bijection to and define as the free module with basis , formed by all linear combinations

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

If is a finite set with elements, this module is after choosing an ordering of .

See also

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

Portions of 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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Free Module

Cite this as:

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

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