Specifically, given a sequence of cohomology groups Hk(X;R) on X with coefficients in a commutative ringR (typically R is Zn, Z, Q, R, or C) one can define the cup product, which takes the form
The cup product gives a multiplication on the direct sum of the cohomology groups
This multiplication turns H•(X;R) into a ring. In fact, it is naturally an N-graded ring with the nonnegative integer k serving as the degree. The cup product respects this grading.
The cohomology ring is graded-commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree k and ℓ; we have
A numerical invariant derived from the cohomology ring is the cup-length, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example a complex projective space has cup-length equal to its complex dimension.