You just need to show that a field can't contain zero-divisors. Since a ring that isn't an integral domain has zero divisor by definition, and if a ring is contained in another ring they have the same multiplication, addition, and zero, a non-integral domain can't be contained in a field.
--Dfreidin 17:26, 29 October 2008 (UTC)
This was very helpful. Thanks a lot.
I also found the information on page 250 helpful (the definition of a field and the information below it).
