There is an exercise (chapter 3, Exercise 10, page number 44) in atiyah mc Donald.
$A$ is absolutely flat iff $A_m$ is a field for each maximal ideal $m$.
I actually proved this statement and it is not hard to do. But my question is where will maximal ideals map to?
We have the following, "There is an one-to-one correspondence between prime ideals of $A_m$ and prime ideals of $A$ not intersecting complement of $m$."
In the light of the above fact, in localization of absolutely flat ring at maximal ideal gives field, which has only two ideals. Now $0$ in $A$ corresponds to $0$ in $A_m$. So, To which ideal in $A_m$, $m$ corresponds to?
This might be some kinda trivial or silly question. But I was clearly missing something and I couldn't figured it out. Please help.
Added: Assume $A$ is an integral domain, if necessary. So that $0$ will be prime.
If $A$ is an absolutely flat integral domain, then $A$ is a field, this is because absolutely flat rings are zero-dimensional, so every prime ideal is maximal. Thus there will be no $m$ to consider as $0$ is the only prime ideal in $A$.