I'm trying to understand the result for the de Rham cohomology ring of flag manifolds in Differential Forms in Algebraic Topology by Bott and Tu. I'm sort of starting from the result and working backwards to pick up what I need, and I thought I had and understanding of the result in hand, but this point is confusing me. At the bottom of pg. 284 they have the result for flag bundles:
$$ H^* (Fl(E)) = H^* (M)[x_1,...,x_n]\diagup \left( \prod_{i=1}^n (1 + x_i ) = c(E) \right) $$
and immediately below this they specialize to a flag manifold:
$$ H^* (Fl(V)) = \mathbb{R} [x_1 , ... , x_n] \diagup \left( \prod_{i=1}^{n} (1 + x_i) = 1 \right) .$$
I think their notation is confusing me slightly. They use parenthesis for the ideal generated by an element. So in the flag manifold expression, the Chern classes are all $0$, hence the total Chern class is $1$, and we're quotienting by the ideal generated by $1$. But this should be the whole ring, so the cohomology ring should be trivial. This seems unlikely to me, and they keep expressing the result in this seemingly-odd form instead of saying "it's trivial" so I feel as though I'm misunderstanding some key fact and misinterpreting this result. What am I missing here?