TITLE: How to show that a finite commutative ring without zero divisors is a field?
QUESTION [15 upvotes]: $R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field?

I'm just starting with abstract algebra and I'd really appreciate if someone could explain it to me.

REPLY [2 votes]: Another possibility: Let $x\in R$, $x\neq 0$. Since $R$ is finite, there must be some repetition in the infinite list $x^0, x^1, x^2, x^3, x^4,\ldots$. Hence there are exponents $i < j$ with $x^i = x^j$. So $x^i(x^{j-i} - 1) = 0$. Since $R$ has no zero divisors, we get $x^{j-i} - 1 = 0$ ($x^i \neq 0$ since $x\neq 0$ and $R$ has no zero divisors). So $x\cdot x^{j-i-1} = 1$, showing that $x$ is invertible.