In this lecture we recall the definition of a ring, and show that if $X$ is a topological space and $R$ is a ring then we can endow the singular cohomology of $X$ with coefficients in $R$ with the structure of a graded ring.

This is the so-called cup product, which starts from the observation that one may “adjoin” a singular $n$-cocycle and a singular $m$-cocycle together to form a singular $(n+m)$-cocycle.

Remark

Why is the cup product called the cup product? Amusingly, as far as I can tell, the name goes back to Whitney, who used the notation $\alpha \smile \beta$ for the cup product, and then, since $\smile$  looks like a cup, decided to call it the "cup product". 🙃