Let $A$ be a Noetherian commutative ring with unit. Let $P$ be a prime ideal. Then according to Atiyah-Macdonald, the height of the ideal $P$ is the maximum of the lengths $n$ of chains $P_0\subsetneq P_1\subsetneq \dots \subsetneq P_n=P$ where $P_i$ are prime ideals.
I have been told that it is obvious that the height of a prime ideal $P$ is zero if and only if it is a minimal prime ideal. I fail to understand this. To me, it seems that any nonzero ideal must have by definition height at least $1$, since if $P\neq 0$, then we can have $P_0=(0)\subsetneq P_1=P$. Can anyone cast light on my misunderstanding? Am I misinterpreting the definition?
By definition, a minimal prime ideal $P$ is a prime ideal that (with respect to inclusion) it is minimal with that property.
We always have
$$ (0) \subseteq P $$
and so when the ring in consideration is a domain, the zero ideal is prime, and thus the former inclusion proves that the only minimal ideal is the zero ideal.
When our ring is not a domain, the zero ideal is not prime. Therefore $(0) \subseteq P$ is not a chain of prime ideals.
The minimality condition on $P$ tells us precisely that there are no prime ideals $Q$ such that $Q \subsetneq P$, and thus $P$ has length zero.