Every prime ideal is either zero or maximal in a PID.

Question

$(1)$ Let $R$ be a commutative ring with $1\neq 0.$ If $R$ is a PID, show that every prime ideal is either zero or maximal.

In many books I have found the proof of the above statement where they show that

(2)Let $R$ be a commutative ring with $1\neq 0.$ If $R$ is a PID, then every nonzero prime ideal is maximal.

I have revised the question now, how can prove that $(1)$ is true using $(2)$. I am of the opinion that both questions are similar and that's why I am asking this question if I am mistaken then please explain why these two are different. Thanks.

Answer 1

In your revised question, you ask us to show that Every nonzero prime ideal is maximal in a nontrivial PID $\implies$ Every prime ideal is either zero or maximal in a nontrivial PID.

This is trivial. Take a prime ideal. If it's zero, that's fine. If it's not, then the assumed statement says exactly that it's maximal. There's nothing left to show.

Showing the other direction is as trivial, too.

EDIT

The OP asks in a comment:

Sometimes the trivial things are difficult to understand! You said, "If it's zero, that's fine." and my question is that: why is it fine? Everyone says it is trivial but I cannot wrap my brain around it. Can you elaborate and explain it with a microscopic lens?

The statement we want to show is Every prime ideal is either zero or maximal in a nontrivial PID. In other words, if we take a prime ideal, we have to show that it is either zero, or that it is maximal.

1. If it is not zero, then by assumption we know that it is maximal (this is the statement from (2) ).

2. If it is zero, than we don't care. Why don't we care? Because we wanted to show that prime ideals are either zero or maximal. So we have shown that all nonzero prime ideals are maximal, and the zero ideal is in fact the zero ideal. That's why I can say that "The zero ideal is zero, and that's fine." There is nothing to prove about that case.

Answer 2

Hint $\ $ By definition $\rm\ [P\ne 0\:\Rightarrow\: P\ max]\iff [P=0\ \ or\ \ P\ max] $

Recall that, by definition: $\rm\ [A\Rightarrow B]\iff [\lnot A\ \ or\ \ B]$