Basis of Free Module is No Greater than Generator
Theorem
Let $R$ be a commutative ring with unity.
Let $M$ be a free $R$-module with basis $B$.
Let $S$ be a generating set for $M$.
Then:
- $\size B \le \size S$.
That is, there exists an injection from $B$ to $S$.
Outline of Proof
Because $S$ is a spanning set, we can construct a surjective homomorphism $R^{\paren S} \to R^{\paren B}$.
Using Krull's Theorem we divide through by a maximal ideal of $R$ to reduce it to the case where $R$ is a division ring, that is, the case of vector spaces.
We then conclude by comparing cardinalities and using Basis of Vector Space Injects into Generator.
Proof
Because $S$ is a generating set, there is a surjective homomorphism
- $\phi : R^{\paren S} \to M$
where $R^{\paren S}$ is the free $R$-module on $S$.
Because $B$ is a basis, there is an isomorphism
- $\psi : R^{\paren B} \to M$
Thus $f = \psi^{-1} \circ \phi: R^{\paren S} \to R^{\paren B}$ is a surjective module homomorphism.
By Krull's Theorem, there exists a maximal ideal $M \subset R$.
By Maximal Ideal iff Quotient Ring is Field, $R / M$ is a field.
Let $k = R / M$.
Let $\pi: R \to k$ denote the quotient mapping.
By some universal properties, there exists a $k$-module homomorphism $\bar f: k^{\paren S} \to k^{\paren B}$ such that:
- $\pi^B \circ f = \bar f \circ \pi^S$
Where $\pi^B$ denotes direct sum of module homomorphisms.
Because $\pi^B \circ f$ is surjective, so is $\bar f$.
Let:
- $C$ be the canonical basis of $k^{\paren S}$
- $D$ be the canonical basis of $k^{\paren S}$
Because $C$ is a generator of $k^{\paren S}$ and $\bar f$ is surjective, $\map {\bar f} C$ is a generator of $k^{\paren B}$.
We have:
| \(\ds \size B\) | \(=\) | \(\ds \size D\) | Cardinality of Canonical Basis of Free Module on Set | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \size {\map {\bar f} C}\) | Basis of Vector Space Injects into Generator | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \size C\) | Cardinality of Image of Mapping not greater than Cardinality of Domain | |||||||||||
| \(\ds \) | \(=\) | \(\ds \size S\) | Cardinality of Canonical Basis of Free Module on Set |
$\blacksquare$
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