First, a ring does not necessarily have a multiplicative identity- though those that do are important enough that 'rings with identity" is a area of study in itself. The crucial difference is that a field not only has a multiplicative identity but every member, except the additive identity, 0, has a multiplicative inverse. The set of all integers is a ring in which only the multiplicative identity, 1, has a multiplicative inverse. The set or rational numbers is a field. (The set of even integers is a ring that does not have a multiplicative identity.)
