1. The problem statement, all variables and given/known data Prove that the group Q/Z under addition cannot be isomorphic to the additive group of a commutative ring with a unit element, where Q is the field of rationals and Z is the ring of integers. 2. Relevant equations The tools available are introductory-level group theory and ring theory, from a first course in Abstract Algebra. 3. The attempt at a solution I was thinking that it might be helpful to show that Q/Z has no unit element (since 1 is in Z), and then show that if this were true, then Q/Z must have a unit element. However, I'm not quite sure how to get started, or if I'm even taking a correct approach.