Free module

In mathematics, a free module is a module having a free basis.

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Mathematics - Module

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

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• 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;
• if r₁e₁ + r₂e₂ + ... + r_ne_n = 0, then r₁ = r₂ = ... = r_n = 0 (where 0 is the zero element of M and 0 is the zero 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.

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

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The definition of an infinite free basis is similar, except that E will have infinitely many elements. However the sum must be finite, and thus for any particular x only finitely many of the elements of E are involved.

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

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