Cohomology ring of the connected sum of two complex projective planes
= Cohomology ring of the connected sum of two complex projective planes
{title2=$\mathbb Q[a_2,b_2]/(ab,a^2-b^2)$}
For the standard complex orientations, the two summand classes have zero product and equal squares, the oriented degree-four class. The quadratic relations imply all cubic monomials vanish. The rational Betti numbers are $1,2,1$ in degrees zero, two and four.