I have an example sheet that asks me to compute the cohomology rings for two spaces, say X and Y, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_d$ respectively. It then asks whether X and Y are homotopy equivalent. Now I know that if two spaces have different cohomology rings then they can't be homotopy equivalent.
But if we have the cohomology ring of X with one set of coefficients and Y with another can we compare them in any useful way? Perhaps by using tensor product to restrict scalars of the $\mathbb{Z}$ ring?
The other possibility of course is that the question was misphrased and I am meant to do it with both sets of coefficients and compare like with like. Either way I would like to know whether comparisons can be made.
I am mainly asking whether there is any general/ commonly used technique of comparing spaces via their cohomology rings even if the coefficients are different in each. This would be particularly useful if say the rings of the spaces are easier to compute for certain coefficients.
Perhaps the Universal Coefficient Theorem is what you are looking for? A continuous map $f \colon X \to Y$ induces a map $f^*$ from the Universal coefficient short exact sequence for $Y$ to the one for $X$. That could be of some help.