I just had this thought when I was thinking how I was introduced to the concept of number in primary school and I came upon the conclusion that the numbers we were taught to manipulate (adding, multiplying, long division), i.e. the rational numbers that have a finite number of digits after the decimal point. It is not difficult to see that they form a ring. So I tried to find a way to find out what its ideals would look like. First of all I would look at principal ideals. When I plug out the decimal point and factor the resulting integer what would I obtain: the integer primes! (with some exceptions). On the other hand any prime $p$ generates a prime princpal ideal. The only exceptions are $2$ and $5$ which turn out to be units. Why is this ring nowhere given as an example, and what other remarkable properties does it have?


That ring is (isomorphic to) the localization of $\mathbb Z$ at the multiplicative set $S=\{2^i5^j:i,j\geq0\}$, and as such it is a standard example!

  • $\begingroup$ I'm not into localizations already, but thanks for this nice remark. $\endgroup$ – Marc Bogaerts Oct 22 '14 at 18:38
  • $\begingroup$ if we take $i=j$ is also enough, I guess. $\endgroup$ – mesel Oct 22 '14 at 18:39

As Mariano pointed out, it is the localization $\mathbb{Z}[10^{-1}]$ of the PID $\mathbb{Z}$ at the element $10$. Hence it is a PID again. Its prime elements are those which haven't been inverted, i.e. all prime numbers except for $2$ and $5$. Since we have a PID, the ideals are then easy to determine; these are the principal ideals generated by natural numbers which are either zero or (positive and not divisible by $10$).

We may also understand commutative rings via their spectrum. We get $\mathrm{Spec}(\mathbb{Z}[10^{-1}])$ from $\mathrm{Spec}(\mathbb{Z})$ by removing the two closed points $(2)$ and $(5)$.

$$\stackrel{(0)}{\Large\bigcirc} {~~\large \leadsto} ~~~ \stackrel{(3)}{\bullet} ~ \stackrel{(7)}{\bullet} ~ \stackrel{(11)}{\bullet} ~ \stackrel{(13)}{\bullet} ~ \stackrel{(17)}{\bullet} \dotsc$$

The generic point $(0)$ specializes to the closed points $(p)$, $p \in \mathbb{P} \setminus \{2,5\}$.


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