Let $(A,\mathfrak{m})$ be a local Artin ring. Then the following statements are equivalent:

1. Every ideal in $A$ is 1;;principal
2. The maximal ideal $\mathfrak{m}$ is 11;;principal
3. $di{m}_k(m/m^2)\le 1$, where $k=A/\mathfrak{m}$, and dimension refers to vector space dimension.

Proof:
To show 3 implies 1, we apply Lemma 9.2 - Nakayama Lemma for when $di{m}_k(m/m^2)=0$, and for the other case show that every ideal is 1;;a power of $m^i$.