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Hatcher doesn't really say anything about defining the cup product for cellular cohomology, instead he gives the definition in terms of singular and simplicial cohomology, and my interpretation is that his definition would not generalize in a nice way. Therefore, I was wondering if/how higher dimensional cells affect the cup product?

Assume that we pick an element $\alpha\in H^k(X)$ and $\beta\in H^l(X)$. Then the product would yield an element in $H^{k+l}(X)$. If we would restrict ourselves to the $X^{k+l+1}$-skeleta of $X$, then my guess is that the corresponding elements in $H^*(X^{k+l+1})$ would multiply in the exact same way?

If this holds, then from computing the cohomology ring of $\mathbb{C}\textrm{P}^\infty$, this would immediately provide the cohomology ring of $\mathbb{C}\textrm{P}^n$ by just cutting of the polynomial ring.

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Your guess is correct. This follows immediately from functoriality of cohomology rings and the fact that restriction $H^i(X)\to H^i(X^{k+l+1})$ is an isomorphism for $i\leq k+l$.

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