nLab
cohomology ring

Similar to the discussion at cohomology group, a cohomology set $H_{*}(X,A)$ may have the strcuture of a ring if the coefficient object $A$ has suitable properties.

In particular for $H=$Top, $H_{*}(X,A)$ is a ring when $A$ is an $E_{\mathrm{\infty }}$-ring.