Free Module is Isomorphic to Free Module on Set
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Theorem
Let $M$ be a unitary $R$-module.
Let $\BB = \family {b_i}_{i \mathop \in I}$ be a family of elements of $M$.
Let $\Psi: R^{\paren I} \to M$ be the morphism given by Universal Property of Free Module on Set.
Then the following are equivalent:
- $\BB$ is a basis of $M$
- $\Psi$ is an isomorphism
Proof
Follows directly from:
- Characterisation of Linearly Independent Set through Free Module Indexed by Set
- Characterisation of Spanning Set through Free Module Indexed by Set.
$\blacksquare$