maximal ideal of PIR

Question

PIR means principal ideal ring.This is extended concept of PID, we do not assume integral domain.

Is it true that every nonzero prime ideal is maximal, even in PIR ? Or, do we need to restrict the condition ?（For example, non nilpotent prime ideal） Thank you for your help.

Answer 1

I think the best one can say is that the class of commutative rings for which the nonzero prime ideals are maximal can be classified in two pieces:

1. domains of Krull dimension $1$ (which come in a large variety and can't really be described more simply)
2. rings of Krull dimension $0$ (also called $\pi$-regular rings) which can be characterized as
   1. Rings for which chains of the form $(x)\supseteq (x^2)\supseteq (x^3)\supseteq\ldots$ stabilize for all $x$
   2. $R/N$ is von Neumann regular, where $N$ is the nilradical of $R$.

Principal ideal domains fall under the first item, but since taking finite ring products increases the Krull dimension but preserves the property of being a PIR, you will quickly leave the class (as mentioned in the comments.)

Any Artinian PIR is zero dimensional, though.