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?

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

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