Def. A ring R is called a pm ring if each prime ideal is contained in exactly one maximal ideal.
I asked AI to give some examples of pm rings. The answer is:
Examples of PM rings in algebra include:
The ring of integers, ℤ.
The ring of polynomials with coefficients in a field, F[x]....
But in either case $0$ is a prime ideal that is contained in infinitely many maximal ideals. Am I missing something?
In this article, PM-rings are defined as each prime ideal is contained in a unique maximal ideal,(1) and examples are given. Notably $\mathbb Z$ is not there, and you can it would be there if valid, so I'm fairly certain this is just AI blabber. The other example it gave is also wrong.
On the other hand, I have seen the condition every nonzero prime is contained in a unique maximal ideal (2) for example in the definition of h-local domains.. Of course, they match when the ring isn't a domain...
Why on earth would you ask AI math questions? It is highly unreliable beyond asking for common definitions. I once asked it for an example of a division ring not isomorphic to its opposite ring and it said "the quaternions, because the opposite ring of the quaternions is a commutative ring..."
Actual examples would include local rings and zero dimensional rings, and also quotients of $h$-local domains by an ideal other than zero.