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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?

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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!

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  • $\begingroup$ 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
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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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