Why is $S/R$ a ring extension?

Question

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.

Answer 1

I have no sources for any of this, but I think the reason is rather low level. The "/" is just shorthand for "over". For example I often abbreviate "a vector space over $k$" to "a v.s./k" in my notes. Same for "module over a ring" so seems reasonable to expect the notation to be used for extensions of rings as well.

Answer 2

I don't think there's anything very deep here, it's just another confusing bit of notation.

I don't actually know the history, but it might be a generalization of the same notation specifically for field extensions. Since quotients of fields are not interesting, there's little scope for confusion there.

In any case, this shouldn't be too confusing, since for $S/R$ to be a quotient $R$ has to be an ideal, and for it to be an extension $R$ has to be a subring, and the only thing that is both is the rather trivial case $R=S$. If you allow rings without $1$ then this is no longer true, and this notation should be avoided.