Confusion about coordinate ring of a fiber. I am reading an example given on page 82 in Eisenbud and Harris's The Geometry of Schemes, and I am confused about some terminology. Let $A:=\mathbb{Z}[\sqrt{3}]$, the ring of integers for $\mathbb{Q}(\sqrt{3})$. The natural inclusion $\mathbb{Z}\hookrightarrow A$ induces a map $\operatorname{Spec}A\rightarrow\operatorname{Spec}\mathbb{Z}$. Looking at the fiber of a point $[(p)]\in\operatorname{Spec}\mathbb{Z}$ is equivalent to examining the primes lying above $p$ in $A$. In the case that $p=2$ or $p=3$, we get $2A=(1+\sqrt{3})$ and $3A=(\sqrt{3})^2$, respectively, and the residue fields at the points $(1+\sqrt{3})$ and $(\sqrt{3})$ are $\mathbb{F}_2$ and $\mathbb{F}_3$, respectively.
Now, for these cases, the authors say that for $p = 2$ or $3$, the fiber over $(p)$ is a "single, nonreduced point, with coordinate ring isomorphic to $A/\mathfrak{p}^2$".
My question is, how is the "coordinate ring" here defined? I know when $A$ is a reduced, finitely generated $K$-algebra over a algebraically closed field $K$ that $A$ is itself the coordinate ring. But I'm confused as to what it means here precisely, and can't find any definition given in the book. Can someone explain to me what definition is being used?

If $f:X\to Y$ is a morphism of schemes and $y\in Y$ is a point, the fiber over $y$ is $\operatorname{Spec} \kappa(y)\times_Y X$ (see here for a proof that the underlying topological space of this scheme is really the same as the set $f^{-1}(y)$ with the subspace topology). When $X=\operatorname{Spec} A$ and $Y=\operatorname{Spec} R$ are affine, like in your problem, $\operatorname{Spec} \kappa(y)\times_Y X$ is also affine: it's the spectrum of $\kappa(y)\otimes_R A$. The ring $\kappa(y)\otimes_R A$ is the coordinate ring of the fiber.
