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  "body": "\n## Definition\n\nLet $X$ be a topological space and let $R$ be a [[commutative ring|0310REJJY8]] with $1$.\nThe *cohomology ring* of $X$ with coefficients in $R$ is the direct sum of the [[singular cohomology|6N7BC70CX0]] groups\n$$H^\\bullet(X; R) = \\bigoplus_{k \\ge 0} H^k(X; R).$$\nIts multiplication extends the [[cup product|JX47MS1R1H]] $H^k \\times H^l \\to H^{k+l}$ bilinearly to all of $H^\\bullet(X; R)$.\n\n## Remarks\n\nIt is an associative [[graded ring|8KB4SQGC0C]] with unit, and an $R$-algebra.\nIt is graded commutative: $ab = (-1)^{kl} ba$ for $a \\in H^k$ and $b \\in H^l$.\nA continuous map $f \\colon X \\to Y$ induces a ring homomorphism $f^* \\colon H^\\bullet(Y; R) \\to H^\\bullet(X; R)$.\n\nExample: $H^\\bullet(S^n; \\Z) \\cong \\Z[x]/(x^2)$ with $x$ of degree $n \\ge 1$, since $x^2$ would lie in $H^{2n}(S^n) = 0$.\nFor the complex projective space $\\C P^n$, $H^\\bullet(\\C P^n; \\Z) \\cong \\Z[x]/(x^{n+1})$ with $x$ of degree $2$.\n\nNon-example: [[singular homology|M0Q5CFJ7ZH]] $H_\\bullet(X)$ is a graded group of the same shape, but in general it carries no natural product of this kind.\nThe cup product needs the diagonal map $X \\to X \\times X$, which goes the wrong way for homology.\n",
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