\begin{definition}[Definition:Characteristic Polynomial of Element of Algebra]
Let $A$ be a commutative ring with unity.
Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$.
Let $b \in B$.
The '''characteristic polynomial''' of $b$ is the characteristic polynomial of the regular representation $\lambda_b : B \to B$ over $A$.
\end{definition}