Localizations of $\mathbb{Z} / (30)$

Question

I am trying to find all localizations of the ring $\mathbb{Z} / (30)$. To do this, I know that I need to find multiplicatively closed subsets of $\mathbb{Z} / (30)$, and then the localizations are of the form $W^{-1} \mathbb{Z} / (30)$, where $W$ is a multiplicatively closed subset. But is there any systematic way of finding all of the multiplicatively closed subsets? I can't seem to find a way of doing this. Also, as I am only looking for localizations up to isomorphism, do I need all of the multiplicatively closed subsets, or only some of them?

Answer 1

It is enough to consider the subsets of $\{2,3,5\}$.

Basically, if the multiplicative subset contains only units, then the localization is nothing but the ring itself. So you may think that basically we need to specify which non units we want to be inverted. Nonunits in $\mathbb{Z}/(30)$ are $0$ and the class of integers which are divisible by $2,3$ or $5$. Hence it is enough to specify which subset of $\{2,3,5\}$ you want to invert (i.e. take the subset and set $W$ to be the multiplicative set spanned by this subset). There are $8$ of these subsets.