The Leray-Hirsch theorem: let $k$ be a field. Given a fibration $F \to E \to B $ with $F, B$ path connected and suppose system of local coefficient is zero and the following condition satisfied (a) $H^n(B;k) $ is finitely dimensional for each $n$ and (b) $ i^*:H^*(E;k) \to H^*(F;k)$ is onto, here $i$ is the inclusion of fiber. Then $$H^*(E;k) \cong H^*(B;k)\otimes_k H^*(F;k) $$ as vector spaces.
My question is: What are some extra condition do we need so that above becomes ring isomorphism. In particular for the field $k=\Bbb Z/2 \Bbb Z$ what extra condition do we need.
Thank you very much for your kind help.