I'm interested in the following problem:
Let $X$ and $Y$ be finite CW complexes and $R$ and $S$ rings. Suppose you are given the cohomology rings $H^* (X; R)$ and $H^* (Y; S)$. Is there an easy way to compute $H^* (X \times Y; R \times S)$?
The problem I saw had $X = K^2$ the Klein bottle and $R = \mathbb{Z}/2\mathbb{Z}$; and $Y = L(3;3) = S^3 / \mathbb{F}_3$ the three-fold lens space and $S = \mathbb{Z}/3\mathbb{Z}$.
I was attempting to use a combination of the Kunneth formula and the Universal Coefficients Theorem, but I was getting stuck on finding the ring structure.
Thank you!
