Why is the largest invariant factor the minimal polynomial, and why is the product of invariant factors the characteristic polynomial?

Question

I'm just learning the primary decomposition theorem for finitely generated modules over a PID, and its application to linear algebra. Let $V$ be a vector space over $K$, and let $T: V \to V$ be a $K$-linear operator. We can then view $V$ as a $K[T]$-module, and we have that $$ V \cong K[T]/(f_1(T)) \oplus \cdots \oplus K[T]/(f_n(T)), $$ where $f_i$ divides $f_{i+1}$. Assume the $f_i$ are monic. My understanding is that $f_n$ is $T$'s minimal polynomial, and $f_1 f_2 \cdots f_n$ is $T$'s characteristic polynomial. How come this is the case?