Local ring of Krull dimension zero

Question

In commutative rings text books it is usually asked to prove that as long as

$(R,m)$ is a Noetherian local ring, the following are equivalent:
(i) $m^n=m^{n+1}$ for some integer $n$;
(ii) $m^n=0$ for some $n$;
(iii) the Krull dimension of $R$ is $0$.

By Nakayama lemma I found that (i) yields (ii), the reverse direction being evident. But, their relevance to (iii) is unclear to me. By the way, is the Noetherianness a necessary condition? Thanks for any help.

Answer 1

Patrick's answer shows that Noetherianness is sufficient, and I hope to show it's necessary by providing some counterexamples.

Clearly (ii) always implies (i) and (iii) even without the Noetherian condition.

Example 1: Local, Krull dimension $0$, but $m^k\neq \{0\}$ for all $k$.

Take a field $F$ and the polynomial ring $F[x_2,x_3,x_4\ldots]$ in countably many variables, and take the quotient by the ideal $(x_2^2,x_3^3,x_4^4,\ldots)$. As you can see $m=(x_2,x_3,x_4,\ldots)$ is not finitely generated and nil, and it contains elements of arbitrarily high nilpotency index, so it is not nilpotent.

Example 2: Local and $\{0\}\neq m=m^2$ .

In the same picture as above, throw in more generators to the ideal you are quotienting by to include things of the form $x_i -x_{i+1}x_{i+2}$. When this is done, $m=\{x_2,x_3,x_4,\ldots\}$ is now idempotent in addition to being the unique maximal ideal.

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(Yeah, I see that example two actually does both things, but the thing is that I came up with the first one and then tinkered a long time before settling on the second one :) )

Answer 2

(ii) $\Rightarrow$ (iii)

Suppose $\mathfrak p \subsetneq \mathfrak m$. Then $\mathfrak p^n \subsetneq \mathfrak m^n$, for if we had equality, $\mathfrak m^n \subseteq \mathfrak p$, and since $\mathfrak p$ is prime, we would have $\mathfrak m \subseteq \mathfrak p \subseteq \mathfrak m$. But by (ii) $\mathfrak m^n = 0$, so $R$ must have Krull dimension $0$.

(iii) $\Rightarrow$ (i)

Since $A$ is Noetherian (this is where we use it) and every prime ideal is maximal, $A$ is Artinian. This means the descending chain $$ \mathfrak m \supseteq \mathfrak m^2 \supseteq \cdots \supseteq \mathfrak m^n \supseteq \cdots $$ must be stationary.

(If you need more details as to why $A$ is Artinian, I can send you my commutative algebra notes which contains all the details in a very concise manner.)

Hope that helps,