# Example of PID and its maximal ideal

• tsang
In summary, the conversation discusses an example involving PID and maximal ideal, where a subring of the rational numbers is shown to be a PID with only one maximal ideal. The reasoning for this is that the more units there are in the subring, the fewer ideals there are, as any ideal containing a unit is the whole ring. This example is based on starting with a PID and adding in more denominators, resulting in a smaller number of ideals that are still generated by the same principal generators. Additionally, the conversation mentions that anytime the set of non-units forms an ideal, it contains all other ideals.
tsang
I got one example on my notes about PID and maximal ideal. I feel it is a strange example as it doesn't make sense to me, and there are no explanations. It says:

For a prime p$\in$$\mathbb{N}$, denote by $\mathbb{Z}_{(p)}$ the subring of $\mathbb{Q}$ given by
$\mathbb{Z}_{(p)}$={$\frac{m}{n} \in$$\mathbb{Q}$|p does not divide n}.
Then $\mathbb{Z}_{(p)}$ is a PID, and it has exactly one maximal ideal.

I can't see the reason of this example at all, and I'm not able to imagine what are the ideals like in Z_(p), can anyone please explain to me why it is a PID and only has one maximal ideal? Thanks a lot.

Show that all ideals are of the form

$$\{\frac{m}{n}\in \mathbb{Z}_{(p)}~\vert~m\in I\}$$

for I an ideal in $\mathbb{Z}$.

intuitively, the more units there are, i.e. the more denominators are allowed, the fewer ideals there are, since any ideal containing a unit is the whole ring.

so you started from a pid, namely Z, and added in a lot of denominators, i.e. anything not divisible by p, so you lost a lot of ideals, and the remaining ones are probably still generated by the same principal generators as before.

i.e. try intersecting an ideal of your ring with Z, and see if the generator of that ideal also generates your original ideal.

anytime the set of non units itself forms an ideal, that ideal contains all others. what are the non units here?

## 1. What is a PID?

A PID, or a principal ideal domain, is an algebraic structure in abstract algebra that satisfies certain properties. It is a commutative ring with a multiplicative identity where every ideal is principal, meaning that it can be generated by a single element.

## 2. What is an example of a PID?

An example of a PID is the ring of integers, Z. Every ideal in Z is generated by a single integer, making it a principal ideal domain.

## 3. What is the maximal ideal of a PID?

The maximal ideal of a PID is the ideal that is not contained in any other proper ideal. In other words, it is the largest possible ideal in the PID. In a PID, the maximal ideal is always a prime ideal.

## 4. How is a maximal ideal different from other ideals in a PID?

A maximal ideal is different from other ideals in a PID because it cannot be properly contained in any other ideal. This means that it cannot be generated by a single element, unlike other ideals in a PID. Additionally, the quotient ring formed by dividing a PID by its maximal ideal is a field, which is not true for other ideals in a PID.

## 5. What is the significance of maximal ideals in a PID?

Maximal ideals play a crucial role in the structure of a PID. They are used to define the concept of prime elements, which are essential in unique factorization of elements in a PID. Additionally, maximal ideals are used in the study of algebraic number theory and algebraic geometry.

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