Cohomology ring

Summary

The cohomology ring of a space $X$ with coefficients in a commutative ring $R$ is the direct sum $H^\bullet(X; R) = \bigoplus_{k \ge 0} H^k(X; R)$ of its singular cohomology groups. Its multiplication is the cup product. It is a graded commutative ring and a homotopy invariant of $X$. It separates spaces that have isomorphic homology groups, and it is the setting for characteristic classes and cohomology operations.