# $\mathbb{Q}/\mathbb{Z}$ is a torsion $\mathbb{Z}$-module

I'm reading through some chapters of Dummit and Foote Abstract Algebra book and in one of the examples from Projective Modules section they state that $$\mathbb{Q}/\mathbb{Z}$$ is a torsion $$\mathbb{Z}$$-module. So if I understand the quotient modules correctly, to show that $$\mathbb{Q}/\mathbb{Z}$$ is a torsion $$\mathbb{Z}$$-module we need to show that for every element $$a$$ in $$\mathbb{Q}$$ there exists some non-zero element $$r$$ in $$\mathbb{Z}$$ s.t. $$ra \in \mathbb{Z}$$. But we know that every $$a$$ can be written as $$x/y$$, where $$x,y \in \mathbb{Z}$$ thus we can always take $$r=y$$ and we can then conclude that indeed every $$\mathbb{Q}/\mathbb{Z}$$ is a torsion $$\mathbb{Z}$$-module. Is that correct?

• Yes, that is correct – Max Feb 11 at 10:49
• Great, thanks @Max – amator2357 Feb 11 at 10:51
• Your argument is correct. – Dbchatto67 Feb 11 at 11:02