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?

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

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