In the chapter on Artinian rings in "Introduction to Commutative Algebra" by Atiyah and MacDonald, we have:
Proposition 8.6. Let $(A,\mathfrak{m})$ be a local Noetherian ring. Then exactly one of the following holds:
$\frak{m}^n\neq\frak{m}^{n+1}$ for all $n \in \mathbb{N}$;
$\frak{m}$ is a nilpotent ideal, in which case, $A$ is Artinian.
Proposition 8.8. Let $(A,\mathfrak{m})$ be a local Artinian ring. Then the following are equivalent:
$A$ is a principal ideal ring;
$\frak{m}$ is principal;
$\dim_K(\frak{m}/\frak{m}^2)\leq 1$ (where $K=A/\frak{m}$ is the residue field).
In the proof of 8.8, A&M quickly boil down to the case:
$$\dim_K(\mathfrak{m}/\mathfrak{m}^2)= 1 \Rightarrow A \text{ is a principal ideal ring.}$$
A&M begin by explaining that $\mathfrak{m}$ is nilpotent, which is because $\mathfrak{m}$ equals the Jacobson radical of $A$ (as $A$ is a local ring) and the Jacobson radical of an Artinian ring is nilpotent (as A&M prove earlier).
However, doesn't this also follow from 8.6 as Artinian rings are Noetherian (which A&M prove in Proposition 8.5)?
It follows as long as you know that an Artinian ring is Noetherian; I assume they're trying to avoid that fact here.