Quotient of an affine scheme

Question

Let $A$ be a ring and let $X:=\mathsf{Spec} \: A$. If $R$ is a n equivalence relation over $X$, then is the quotient $X/R$ again the spectrum of some ring? If in general it is not, what are the assumptions I need to do about $R$ in order to get $X/R=\mathsf{Spec} \: B$ for some ring $B$?

Answer 1

Here is a nice class of example : if $G$ is a finite group and $X$ affine then $X/G = \text{Spec}(\Bbb C[X]^G)$.

In fact, it is almost accidental, usually for a "nice" algebraic group (i.e linearly reductive) and an linear action $G \times X \to X$ where $X$ is a vector space, then $\text{Spec}(\Bbb C[X]^G)$ will parametrize the closed orbits in $X$. More precisely there is a map $X \to X//G := \text{Spec}(\Bbb C[X]^G)$ coming from the inclusion $\Bbb C[X]^G \subset \Bbb C[X]$, and there is a unique closed orbit in each fiber of this map (for this see Mukai, Introduction to Invariants and Moduli, chapter 4).

This shows that there is very little hope to expect something more general to hold true.