I would like some help with geometrically interpret quotient rings using algebraic geometry. To explain my current understanding using an example, take:
$$ R=\mathbb{C} \left[ x, y, z \right] / \left( xy, yz \right) $$
It seems to be that if you mod out $xy$ so $xy=0$, then $x=0$ or $y=0$. Thus the algebraic subset of $\mathbb{C}^3$ corresponding to the ideal $\left( xy \right)$ is the union of the $x$ and $y$ axes.
Similarly, when we mod out the other equation, we are left with the union of the $y$ and $z$ axes.
What, then, does the ideal $\left( xy, yz \right) $ correspond to? The $x$, $y$, $z$ axes? Is it only the $y$ axis?
Is there some way to geometrically interpret $R$ overall, using similar reasoning in algebraic geometry?
This is where things get a little fuzzy. Any help in finishing off/clarifying the geometric interpretation of $R$ here would be great. Any advice/comments on how to interpret things like this in general would be welcome too
There is a bijection between coordinate rings and affine varieties. In our case, that variety is $V= V(xy, yz)$. A point $x \in \Bbb C^3$ is in $V$ iff either its $y$-coordinate is zero or both its $x$- and $z$-coordinates are zero.
We can write $V = V(x, z) \cup V(y)$. Therefore $V$ is the union of the $xz$-plane and the $y$-axis.