a finite integral domain is a field

A finite integral domain is a field.

Proof:
Let $R$ be a finite integral domain. Let $a$ be nonzero element of $R$.

Define a function $\phi :R\to R$ by $\phi (r)=ar$.

Suppose $\phi (r)=\phi (s)$ for some $r,s\in R$. Then $ar=as$, which implies $a(r-s)=0$. Since $a\ne 0$ and $R$ is a cancellation ring, we have $r-s=0$. So $r=s$, and hence $\phi$ is injective.

Since $R$ is finite and $\phi$ is injective, by the pigeonhole principle we see that $\phi$ is also surjective. Thus there exists some $b\in R$ such that $\phi (b)=ab=1_R$, and thus $a$ is a unit.

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

Note:
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