Definition:Characteristic Polynomial of Linear Operator/Definition 2

Definition
Let $A$ be a commutative ring with unity.

Let $M$ be a free module over $A$ of finite rank $n > 0$.

Let $\phi : M \to M$ be a linear operator.

Let $A \sqbrk x$ be the polynomial ring in one variable over $A$.

Let $I_M$ denote the identity mapping on $M$.

Let $M \otimes_A A \sqbrk x$ be the extension of scalars of $M$ to $A \sqbrk x$.

The characteristic polynomial of $\phi$ is the determinant of the linear operator $x I_M - \phi$ on $M \otimes_A A \sqbrk x$.

Also see

 * Equivalence of Definitions of Characteristic Polynomial of Linear Operator