I have the following (maybe simple) question about the cup product structure in the Serre spectral sequence.
Consider a fiber bundle $S^1 \rightarrow E \rightarrow B$ with euler class $e \in H^2(B)$. Writing out the Serre spectral sequence we obtain that the cohomology of $E$ is isomorphic to
$(\text{coker } e \otimes 1) \oplus (\text{ker } e \otimes \alpha)$
where by coker $e$ and ker $e$ I mean the cokernel and kernel of the operation on $H^*(B)$ given by cupping with $e$. (Here, $\alpha$ is the generator of $H^1(S^1)$.)
I know that the above isomorphism is not necessarily an isomorphism of rings. However, I was wondering if (similar to the Leray-Hirsch theorem) one could show that the isomorphism was true on the level of $(\text{coker } e \otimes 1)$-modules?
Thanks!
