Free module

For an R-module M, the set E = {e₁, e₂, ... e_n} is a free basis for M if and only if:

1) E is a generating set for M, that is to say every element of M is a sum of elements of E multiplied by coefficients in R.

2) if r₁e₁ + r₂e₂ + ... + r_ne_n = 0, then r₁ = r₂ = ... = r_n = 0 (where 0 is the identity element of M and 0 is the identity element of R).

If M has a free basis with n elements, then M is said to be free of rank n, or more generally free of finite rank.

Note that an immediate corollary of (2) is that the coefficients in (1) are unique for each x.

The definition of an infinite free basis is similar, except that E will have infinitely many elements. In general, the summation which generates the elements x of M may be infinite but must converge in whatever sense is appropriate for M. For some modules this will mean that it must be a finite sum, and thus that for any particular x only finitely many of the elements of E are involved.

In the case of an infinite basis, the rank of M is the cardinality of E.