Why $S[x]/f(x)$ is used to denote a quotient ring of polynomials instead of $S[x]/f(x)S[x]$?

Question

I'm a bit confused about the notation used for quotient rings.

In the notation for a quotient rings on both sides of the $/$ there is a set. More specifically a ring divided by one of it's ideals.

For arithmetic modulo integer $n$. We use the notation $\mathbb{Z}/n\mathbb{Z}$. Which looks ok to me.

But for arithmetic modulo some arbitrary polynomial $f(x)$ over some ring $S$ we use the notation $S[x]/f(x)$. There $f(x)$ just a single polynomial, not an ideal in itself. Syntax errors tingling in my brain when I see it. Why don't we write $S[x]/f(x) S[x]$ instead?