Cohomology ring of a collapsed projective subspace
= Cohomology ring of a collapsed projective subspace
For $0<k<n$, let $X=\mathbb{CP}^n/\mathbb{CP}^k$ and let $x_i\in H^{2i}(X;\mathbb Z)$ pull back to $x^i\in H^{2i}(\mathbb{CP}^n;\mathbb Z)$. Additively,
$$
H^q(X;\mathbb Z)\cong
\begin{cases}
\mathbb Z,&q=0\text{ or }q=2i\text{ with }k+1\leq i\leq n,\\
0,&\text{otherwise},
\end{cases}
$$
and its positive-degree multiplication is
$$
x_ix_j=\begin{cases}x_{i+j},&i+j\leq n,\\0,&i+j>n.\end{cases}
$$