The main source of inspiration for this question is this excerpt
Recall: An ultrafilter on the set X gives you a maximal ideal in the ring of all real-valued functions, and these are the only prime ideals.
from this MathOverflow answer of Tom Goodwillie. (In the same thread there is an outstanding answer of our friend Georges Elencwajg.)
Let $X$ be an infinite set, for each $x$ in $X$ let $K_x$ be a field, and let $A$ be the product of the $K_x$.
Here is what I'd like to know:
$(1)$ What are the maximal ideals of $A$, and what are the corresponding residue fields?
$(2)$ What are the non-maximal prime ideals of $A$, and what are the corresponding residue domains?
$(3)$ What are the non-prime primary ideals of $A$, and what are the corresponding residue rings?
$(4)$ How can one describe the Zariski topology on the prime spectrum of $A$? In particular, does this topology depend only on $X$, or does it really depend on the $K_x$?
For $f$ in $A$ and $x$ in $X$, write $f(x)$ for the $x$ component of $f$, and $f^{-1}(0)$ for the set of all $x$ in $X$ such that $f(x)=0$.
Here are the few things I believe I know:
$(5)$ To each $x$ in $X$ is attached an obvious maximal ideal $\mathfrak m_x$ with residue field $K_x$: the set of all $f$ vanishing at $x$.
$(6)$ More generally, to each ultrafilter $U$ on $X$ is attached the set, which is in fact a prime ideal, of all $f$ such that $f^{-1}(0)$ is in $U$. Then (5) corresponds to the principal ultrafilters. But I have no idea about the residue domain associated with a non-principal ultrafilter.
I believe that the answer to (2) is that all primes are maximal.
I (perhaps naively) expect all primary ideals to be prime, and thus maximal.
So, as you see, I need tons of help! Thank you very much in advance!