Cayley-Hamilton Theorem/Finitely Generated Modules

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Theorem

Let $A$ be a commutative ring with unity.

Let $M$ be a finitely generated $A$-module.

Let $\mathfrak a$ be an ideal of $A$.

Let $\phi$ be an endomorphism of $M$ such that $\phi \left({M}\right) \subseteq \mathfrak a M$.


Then $\phi$ satisfies an equation of the form:

$\phi^n + a_{n-1} \phi^{n-1} + \cdots + a_1 \phi + a_0 = 0$

with the $a_i \in \mathfrak a$.


Proof

Let $m_1, \ldots, m_n$ be a generating set for $M$.

Then for each $i$, $\phi \left({m_i}\right) \in \mathfrak a M$, say:

$\displaystyle \phi \left({m_i}\right) = \sum_{j \mathop = 1}^n a_j m_j$

for $i = 1, \ldots, n$.



Thus for each $i$:

$(1): \quad \displaystyle \sum_{j \mathop = 1}^n \left[{\delta_{ij} \phi - a_{ij} }\right] m_i = 0$

where $\delta_{ij}$ is the Kronecker delta.

Now let $\Delta$ be the matrix defined as:

$\Delta := \left({\phi \delta_{ij} - a_{ij} }\right)$



Let $\operatorname{adj} \left({\Delta}\right)$ be the adjugate matrix of $\Delta$.

Recall Cramer's Rule:

\(\displaystyle \operatorname{adj} \left({\Delta}\right) \cdot \Delta\) \(=\) \(\displaystyle \Delta \cdot \operatorname{adj} \left({\Delta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \det \left({\Delta}\right) \cdot \mathbf I_n\)



Multiplying through by $\operatorname{adj} \left({\Delta}\right)$ in $(1)$ and applying Cramer's Rule:

$\displaystyle \sum_{j \mathop = 1}^n \det \left({\Delta}\right) m_i = 0$

Therefore $\det \left({\Delta}\right)$ annihilates each $m_i$ and is the zero endomorphism of $M$.

But $\det \left({\phi \delta_{ij} - a_{ij}}\right)$ is a monic polynomial in $\phi$ with coefficients in $\mathfrak a$.

Thus we have an equation of the required form.

$\blacksquare$


Source of Name

This entry was named for Arthur Cayley and William Rowan Hamilton.