# an Artinian integral domain is a field

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

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

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 $a^{n}\neq 0$ (as $R$ is an integral domain), so we have $1=ar$. Thus $a$ is a unit.

Therefore, every Artinian integral domain is a field.

Title an Artinian integral domain is a field AnArtinianIntegralDomainIsAField 2013-03-22 12:49:37 2013-03-22 12:49:37 yark (2760) yark (2760) 13 yark (2760) Theorem msc 16P20 msc 13G05 AFiniteIntegralDomainIsAField