Example of two spaces indisinguishable by their homology modules (with $\mathbb{Z}$ coefficients) but with different cohomology rings I'm running a student seminar on cohomology (for masters students) and would like to motivate the dualisation of homology by talking about cup products. So I'm looking for an example of two spaces $X$ and $Y$ with the same homology modules but different cohomology rings. Are there any nice-ish examples which would be reasonable to talk about in a seminar?
I do already have the example of $X=\mathbb{R}P^n$ and $Y=\vee_{i\leq n}S^i$ with $\mathbb{Z}/2$ coefficients. The problem with this is that these spaces are distinguished by their homology with $\mathbb{Z}$ coefficients. This might still be sufficient motivation for this kind of seminar, but I'd still prefer to have an example of two spaces where you really need the extra ring structure to tell them apart.
Thanks for any help

A similar example to yours is $\mathbb CP^n$ and $\bigvee\{S^i:0<i\leq 2n$ with $i$ even$\}$, as the cohomology ring of $\mathbb CP^n$, with coefficients in $\Bbb Z$, is
$$\mathbb Z[\alpha]/\alpha^{n+1},\text{ } deg(\alpha)=2.$$
