If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. But whenever I actually think about the notation, I find it to be one of the most confusing conventions in algebra. In almost any other context $S/R$ would mean taking a quotient of $S$ by $R$. It seems much more clear to me write let $R \subset S$ be an extension of rings, but I don't see this notation used very frequently.
So I'm wondering if there's some high level reason we use this notation that I'm not seeing. I'm also curious in what context this notation first appeared.