an Artinian integral domain is a field

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

Let aR with a0. Then RaRa2R.

As R is Artinian, there is some n such that anR=an+1R. There exists rR such that an=an+1r, that is, an1=an(ar). But an0 (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