an Artinian integral domain is a field

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

Let $a\in R$ with $a\ne 0$.Then $R\supseteq aR\supseteq a^{2}R\supseteq \mathrm{\cdots }$.

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

Therefore, every Artinian integral domain is a field.