 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,and its positive-degree multiplication is