Find all prime ideals in $\mathbb{Z}/n\mathbb{Z}$ where $n>1$

Question

I want to find all prime ideals in $\mathbb{Z}/n\mathbb{Z}$ where $n>1$.

I think I have to use the following theorem (because they asked me to prove it right before this exercise, which wasn't too complicated):

Let R be a ring and I ⊂ R an ideal. Let φ : R → R/I be the natural residue class homomorphism. Let I ⊂ J be another ideal. Then J is a prime ideal in R if and only if φ(J) is a prime ideal in R/I.

And

Let R be a principal ideal domain and $I\subset R, I\neq(0)$ an ideal. Then every ideal generated by a irreducible element is a prime ideal

An irreducible element in $\mathbb{Z}$ are exactly the prime numbers. I also know that $\mathbb{Z}$ is a principal ideal domain.

I tried supposing that $J=(p)$ is an ideal of $\mathbb{Z}$, and when $p$ prime, it is a prime ideal. Then if $J$ is a superset of $n\mathbb{Z}$, we have that $\phi(J)$ is a prime ideal of $\mathbb{Z}/n\mathbb{Z}$. Is this correct? For what $p$ values does this hold? What if $J$ is not a superset of $n\mathbb{Z}$ as the theorem requires?

Answer 1

Another theorem says the prime ideals of $R/I$ correspond bijectively to the prime ideals of $R$ containing $I$.

In the case of $\mathbf Z/n\mathbf Z$, this means its prime ideals are generated by the congruence classes of the prime divisors of $n$.

If $J=(m)$ does not contain $n$, i.e. if $m$ does not divide $m$, the image of $J$ in $\mathbf Z/n\mathbf Z$ is the ideal $$J\cdot \mathbf Z/n\mathbf Z=(m)\cdot \mathbf Z/n\mathbf Z=(m,n)/(n)=(\gcd(m,n))/(n).$$

Answer 2

An ideal $I$ in a commutative ring $A$ is prime if and only if $A/I$ is an integral domain; $\mathbb Z/n \mathbb Z$ is a finite ring, so a the quotient by a prime ideal has to be a finite integral domain, which is a field.

Now it is not hard to see that (since the subgroups of $\mathbb Z/n \mathbb Z$ are all isomorphic to $\mathbb Z/ d \mathbb Z$ with $d|n$, and these subgroups are also ideals), this is possible if and only if $(\mathbb Z/ n \mathbb Z)/ I \simeq \mathbb Z / p \mathbb Z $ with $p$ prime; you can easily deduce what are the ideals you are looking for.

Answer 3

Hint: to solve this exercise, it is enough to use the following facts:

• an ideal $I\unlhd R$ is prime if and only if $R/I$ is a domain,
• the characteristic of a domain is prime.

Using these, it is enough to find all quotients of ${\mathbf Z}/n{\mathbf Z}$ of prime characteristic. It should be straightforward to check that they are, in fact, domains.