Question

Why zero ring is artinian as well as noetherian?

Answer 1

Let $\{I_n\}_{n\in\mathbb N}$ be ideals of $\{0\}$, such that $I_n\subset I_{n+1}$ ($I_n\supset I_{n+1}$) for every $n$. Since $I_n=I_{n+1}=\{0\}$ for every $n\ge 1$, we obtain that $\{0\}$ is a noetherian (an artinian) ring.