Cohomology ring of $\mathbb RP^n$ with integral coefficient. I know cup product structure on $H^*(\mathbb{R}P^n;\mathbb{Z}_2)= \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$. How to get  $H^*(\mathbb{R}P^n;\mathbb{Z})$ from this? I have two cochain complexes for two coefficient rings.
Now my question is what will be the induced map between these two cochain complexes and what will be  $H^*(\mathbb{R}P^n;\mathbb{Z})$? 

This is a duplicate of this question, the answer to which is given in a comment. See also this website, which was the first thing I found when I searched for "cohomology of projective space".
