Is a set that is an abelian group under addition and a group under multipliation a field? I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a field? 
Cheers Matt

In general, no, because the distributive laws may not hold.
With that and a minor modification, yes. The multiplicative group has to leave out $0$, because $0$ won't be invertible.
So if you have a ring (with identity) such that the nonzero elements are an abelian group, then yes, you have a field.
If the nonzero elements are just a group then you have a division ring.
