# A function $f:X \to Y$ which induces isomorphisms on all homology groups but $X$ and $Y$ have non-isomorphic cohomology rings.

I am trying to find a function $$f:X \to Y$$, where $$X$$ and $$Y$$ are path connected, such that $$f$$ induces isomorphisms on all homology groups, but $$X$$ and $$Y$$ have non-isomorphic cohomology rings. It may be that this isn't possible - and that would be very helpful to know- but I think I have an example.

I've thought of using $$X = \mathbb{R}P^2 \vee S^3$$ and $$Y = \mathbb{R}P^3$$, since these are classic examples of spaces with isomorphic homology groups and non-isomorphic cohomology rings. Then $$f: X \to Y$$ would be the inclusion map on $$\mathbb{R}P^2$$ (thinking of it as a CW-subcomplex of $$\mathbb{R}P^3$$) and the covering map on $$S^3$$.

It seems clear to me that this map will induce isomorphisms on homology groups for $$i=0, 1, 2$$, since for $$X$$ those homology groups depend only on $$\mathbb{R}P^2$$ and the inclusion of a subcomplex plays nicely with homology. However, I'm stuck on showing that $$f_*:H_3(X) \to H_3(Y)$$ is an isomorphism. I think it should be one, but I'm not sure how exactly to justify it.