Let $(A,\mathfrak{m})$ be a noetherian local ring. Then either:

1. For every $n\in \mathbb{N}$, we have ${\mathfrak{m}}^n\ne {\mathfrak{m}}^{n+1}$
2. ${\mathfrak{m}}^n=0$ for some $n\in \mathbb{N}$, and in this case 1;;$A$ is also Artinian.