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 not 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.