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

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?

• Yeah, it's more properly $S[x]/f(x)S[x]$, or some indication that we are taking the quotient by the ideal generated by $f(x)$, like $S[x]/(f(x)),\ S[x]/\langle f(x)\rangle$,... – Jane Doé May 4 at 17:29
• Do we? Surely we use $S[x]/(f(x))$ or $S[x]/\left<f(x)\right>$. – Lord Shark the Unknown May 4 at 17:32
• @LordSharktheUnknown So the extra parenthesis imply the ideal? – Calmarius May 4 at 17:41
• Also note that $\mathbb{Z}/n$ is occasionally found. – Captain Lama May 4 at 17:41
• @Calmarius Yes, so for instance we also have $\Bbb Z[x]/(3,x^2+1)$ etc. – Lord Shark the Unknown May 4 at 17:43