Localising cyclic abelian groups and Q/Z at a prime

By [L1] and [L2], $(\mathbb{Z}\mathrm{/}n\mathbb{Z}{)}_{(p)}\cong {\mathbb{Z}}_{(p)}{\otimes }_{\mathbb{Z}}\mathbb{Z}\mathrm{/}n\mathbb{Z}\cong {\mathbb{Z}}_{(p)}\mathrm{/}n{\mathbb{Z}}_{(p)}$.

If $a=0$, then $p\nmid n$, so $n\mathrm{/}1$ is a unit in ${\mathbb{Z}}_{(p)}$ by [L3]. Hence $n{\mathbb{Z}}_{(p)}={\mathbb{Z}}_{(p)}$, and step 1.1 gives $(\mathbb{Z}\mathrm{/}n\mathbb{Z}{)}_{(p)}=0$.

If $a>0$, write $n=p^{a}m$ with $(m,p)=1$. Then $m\mathrm{/}1$ is a unit in ${\mathbb{Z}}_{(p)}$, so $n{\mathbb{Z}}_{(p)}=p^a{\mathbb{Z}}_{(p)}$. Reduction modulo $p^a$ identifies ${\mathbb{Z}}_{(p)}\mathrm{/}{p}^a{\mathbb{Z}}_{(p)}$ with $\mathbb{Z}\mathrm{/}{p}^a\mathbb{Z}$, so step 1.1 gives $(\mathbb{Z}\mathrm{/}n\mathbb{Z}{)}_{(p)}\cong \mathbb{Z}\mathrm{/}{p}^a\mathbb{Z}$.

For a class $q+\mathbb{Z}\in \mathbb{Q}\mathrm{/}\mathbb{Z}$, if some integer prime to $p$ kills it then [L4] makes it zero in the localisation; this happens exactly for the torsion of order prime to $p$. On the other hand a class of order $p^r$ cannot be killed by any denominator outside $(p)$, so it survives. Therefore $(\mathbb{Q}\mathrm{/}\mathbb{Z}{)}_{(p)}$ is exactly the $p$-primary torsion subgroup.