Abstract Algebra - Properties of Q/Z

[jfiels3]

Homework Statement

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.

Homework Equations

The tools available are introductory-level group theory and ring theory, from a first course in Abstract Algebra.

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.

 

[micromass]

Basically, what they want you to show that you cannot define a multiplication on Q/Z. So, assume that you do have a multiplication (with a unity), try to derive a contradiction.

 

[Hurkyl]

I was thinking that it might be helpful to show that Q/Z has no unit element (since 1 is in Z),

Don't confuse 1 (the element of Z) with 1 (the unit element of a ring).