Applications of cohomology rings

It was recently asked on math.stackexchange ‘what are cohomology rings good for?’. As the answers there show, the ring structure is, in some sense, a ‘finer’ invariant then the group structure and can be used to prove that spaces which have the same cohomology groups, do not have the same homotopy groups (without resorting to higher homotopy, which is hard!)

Question: Show that and do not have the same homotopy type.

Cohomology Ring of Sphere and Projective Plane

I have been trying to learn about cohomology rings. Thanks to math.stackexchange.com I have realised that, at least for the sphere and projective plane it is not too hard – because they only have two non-zero cohomology groups.