Let $c_i$ be the i-th Chern class of vector bundle $F→E→B$. By Leray-Hirsch formula ,we have $H^*(PE)$ isomorphic to $H^*(B) \bigotimes\mathbb{Z}｛x^1,…,x^{n-1}｝$ as $H^*(B)-module$. Where $PE$ is the projectivization of $E$. $x$ is the generator of $H^*(PF)$.
The question is if $c_i=0$ for all $i$,do we have $H^*(PE)$ isomorphic to $H^*(B) \bigotimes\mathbb{Z}｛x^1,…,x^{n-1}｝$ as rings?