a finite integral domain is a field

A finite integral domain is a field.

Let R be a finite integral domain. Let a be nonzero element of R.

Define a functionMathworldPlanetmath φ:RR by φ(r)=ar.

Suppose φ(r)=φ(s) for some r,sR. Then ar=as, which implies a(r-s)=0. Since a0 and R is a cancellation ring, we have r-s=0. So r=s, and hence φ is injectivePlanetmathPlanetmath.

Since R is finite and φ is injective, by the pigeonhole principleMathworldPlanetmath we see that φ is also surjectivePlanetmathPlanetmath. Thus there exists some bR such that φ(b)=ab=1R, and thus a is a unit.

Thus R is a finite division ring. Since it is commutative, it is also a field.

A more general result is that an Artinian integral domain is a field.

Title a finite integral domain is a field
Canonical name AFiniteIntegralDomainIsAField
Date of creation 2013-03-22 12:50:02
Last modified on 2013-03-22 12:50:02
Owner yark (2760)
Last modified by yark (2760)
Numerical id 11
Author yark (2760)
Entry type Theorem
Classification msc 13G05
Related topic FiniteRingHasNoProperOverrings