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.

Title	an Artinian integral domain is a field
Canonical name	AnArtinianIntegralDomainIsAField
Date of creation	2013-03-22 12:49:37
Last modified on	2013-03-22 12:49:37
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	13
Author	yark (2760)
Entry type	Theorem
Classification	msc 16P20
Classification	msc 13G05
Related topic	AFiniteIntegralDomainIsAField