Showing there is no ring whose additive group is isomorphic to $\mathbb{Q}/\mathbb{Z}$ [closed]

Show that there is no commutative ring with the identity whose additive group is isomorphic to $\mathbb{Q}/\mathbb{Z}$.

Solution 1:

Here's how I would organize this: suppose I have a commutative ring $A$ and an isomorphism $f$ of abelian groups from $(A, +)$ (the additive group of $A$) to $\mathbf Q/\mathbf Z$. Assuming that $A$ has a unit element $1 \in A$, we can look at $f(1) \in \mathbf Q/\mathbf Z$. This element has finite order (why?), so there exists some natural number $n$ such that $n \cdot 1 = 0$. [To avoid confusion: this just means "add $1 \in A$ to itself $n$ times".]

Now, does this imply something about every element of $A$? I'll leave a slight gap here for now, but I think the other important thing to notice (and maybe you noticed it at the "why?") is that $\mathbf Q/\mathbf Z$ contains elements of every finite order.