In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication.
Specifically, given a sequence of cohomology groups Hk(X;R) on X with coefficients in a commutative ring R (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 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.