If $v\in V$ and $v=s(T)f(T)v+t(T)g(T)v$ Can we find out any information about Ker $f(T)$ and Ker $g(T)$?

Question

I am trying to prove that if V is a vector space over $F$ and $T: V \rightarrow V $ is a linear map, then V=Ker$f(T)$ $\oplus$ Ker$(g(T))$.

where $f,g$ are coprime polynomials $\in F[x]$ and $f(T)g(T)=0$.

As there exist polynomials $s,t$ such that $sf+tg=1$, this means that if $v\in V$ then $v=s(T)f(T)v+t(T)g(T)v$.

I wasn't sure how to proceed from here.

Answer 1

Let $u = s(T) f(T) v$ and $w = t(T) g(T) v$ with $s$ and $t$ as in your equation. Thus $v = u+w$ where $g(T) u = 0$ and $f(T) w = 0$. Moreover, if $v \in \ker f(T) \cap \ker g(T)$, then $u = w = 0$ so $v = 0$. That says $V = \ker f(T) \oplus \ker g(T)$.