When does a finite ring become a finite field?

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I'm relatively new to ring theory so this is probably a simple question.

Its easy to see that a finite ring that is commutative and has no-zero divisors (i.e. an integral domain) must have multiplicative inverses.

I am wondering if we can rearrange these properties and always get implication or produce finite rings with only 2 of the above properties.

Explicitly my questions are:

1) Does a finite ring that is commutative with multiplicative inverses always have no-zero divisors? (EDIT: this one is pretty easy too, let $xy=0$ and assume $x\neq 0$. then $x^{-1}xy=x^{-1}0$. Hence, $y=0$ as desired.)

2) Is a finite ring that has multiplicative inverses and no zero divisors always commutative?

If these are actually simple exercises, I'd just like a hint to get started with the proof or any counterexamples.

Thanks :)

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2 Answers

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HINT 1) If $\;\rm R\;$ is finite then $\;\rm x\to r\:x\;$ is onto iff 1-1, so $\;\rm R\;$ is a field iff $\;\rm R\;$ is a domain.

2) is Wedderburn's little theorem.

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Des MacHale has a nice and simple answer to the question When is a finite ring a field?

Theorem: Let $(R,+,\cdot)$ be a finite non-zero ring with the property that if $a$ and $b$ in $R$ satisfy $ab=0$, then either $a=0$ or $b=0$. Then $(R,+,\cdot)$ is a field.

MacHale demonstrates that 1) and 2) are particular cases of this theorem. The article is published online by the Irish Mathematical Society here

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