The cohomology of projective unitary groups

The projective unitary group $PU(n)$ is the quotient of the unitary group $U(n)$ by its center $S^{1}=\{e^{i\theta }I_{n};\theta \in \lbrack 0,2\pi ]\}$, where $I_{n}$ is the identity matrix. Combining the Serre spectral sequence of the fibration $PU(n)\rightarrow PU(n)/T$ with the Gysin sequence of the circle bundle $U(n)\rightarrow PU(n)$, we compute the integral cohomology ring of $PU(n)$ using explicit constructed generators, where $T$ is a maximal torus on $PU(n)$.