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  "body": "\n## Summary\n\nThe 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|6N7BC70CX0]] groups.\nIts multiplication is the [[cup product|JX47MS1R1H]].\nIt is a graded commutative ring and a homotopy invariant of $X$.\nIt separates spaces that have isomorphic homology groups, and it is the setting for characteristic classes and cohomology operations.\n\n## Explanations\n\n### L1 human\n\nThe cohomology groups of a space can be added, and with the cup product they can also be multiplied.\nPutting all degrees together gives one ring, much like a ring of polynomials where each variable has a degree.\nFor the complex projective plane, the ring is generated by one class $x$ of degree $2$ with $x^3 = 0$, while $x^2$ is not zero.\nThe space made by joining a $2$-sphere and a $4$-sphere at a point has the same groups, but there the square of the degree $2$ class is zero.\nSo the two spaces are not homotopy equivalent.\n",
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