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), 2.

Abelian group | abstract vector space | basis | cofree module | free | free product | module | ring