Let $R$ be an integral domain, and assume that $R$ is Artinian.

Let $a \in R$ with $a \neq 0$ Then $R \supseteq aR \supseteq a^2R \supseteq \cdots$

As $R$ is Artinian, there is some $n\in\N$ such that $a^nR=a^{n+1}R$ There exists $r \in R$ such that $a^n=a^{n+1}r$ that is, $a^n1=a^n(ar)$ But $a^n \neq 0$ (as $R$ is an integral domain), so we have $1=ar$ Thus $a$ is a unit.

Therefore, every Artinian integral domain is a field.