My question is: Is there any way to calculate the cohomology ring of a finite dimension manifold using the Morse theory?

The cohomology ring structure strongly rely on the following things $C^\ast(X\times Y)\cong C^\ast(X)\otimes C^\ast(Y)$. I cannot figure out how to use intersection to realize the ring structure.

This question originally come from the studying of Floer Homology and wonder there is a way of defining the cohomology ring of Floer homology just as finite dimensional.