I have stumbled on something is apparently a trivial concept, but the difficulty is that I haven't seen this notation before.
Here is the fragment of a text from lecture notes:
Let us call $\rho$ the map from $S$ to $T$. $$ \rho:\underline{a}=\left(a_{1},\ldots,a_{r}\right)\mapsto\left(x_{1}-a_{1},\ldots,x_{r}-a_{r}\right). $$ Another way of thinking about it is that $\rho$ maps to a map: $$ \rho:\underline{a}\mapsto\left\{ \frac{k\left[x_{1},\ldots,x_{r}\right]}{\left(f_{1},\ldots,f_{r}\right)}\overset{x_{i}\mapsto a_{i}}{\longrightarrow}k\right\} $$
[Where $S=\left\{ \underline{a}\in k^{n}\mid f_{1}\left(\underline{a}\right)=\ldots=f_{r}\left(\underline{a}\right)=0\right\}$, $T=\left\{ \textrm{maximal ideals in}\: k\left[x_{1},\ldots x_{r}\right]/\left(f_{1},\ldots,f_{r}\right)\right\}$, and $k$ is an algebraically closed field.]
I understand everything here, except for this item: $$ \frac{k\left[x_{1},\ldots,x_{r}\right]}{\left(f_{1},\ldots,f_{r}\right)} $$
Normally, this kind of notation is used for quotient of a space by a subspace, for example $A/B$ is a quotient space. However assuming that the above is a quotient space, then I'm not quite sure what kind of subspace is $\left(f_{1},\ldots,f_{r}\right)$. Would it be the vector space with $f_{1},\ldots,f_{r}$ as a basis? A bit confused here.