Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let $T_1, \dots , T_n : V \rightarrow V$ be pairwise commuting nilpotent operators on $V$.
(a) Show the composition $T_1 T_2 \cdots T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried: I know that for any $T_i$, $\ker({T_i}^n) = \ker({T_i}^{n+1}) = \cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?