What is this notation $\mathbb{Z}_3[x]_{x^2+1}$?

Question

What is this notation?

$$\mathbb{Z}_3[x]_{x^2+1}$$

I know $\mathbb{Z}_3[x]$ denotes the polynomials with coefficients in $\mathbb{Z}_3$. What does the $x^2+1$ bit denote?

Answer 1

That is the notation used for the localization of the ring $\mathbb Z_3[x]$ at the multiplicative subset $S=\{1,(x^2+1),(x^2+1)^2,(x^2+1)^3,\dots\}$.

That is, $\mathbb Z_3[x]_{x^2+1}=S^{-1}(\mathbb Z_3[x])$.