Principal ideals in rings.

Question

Consider the following list of principal ideals (2), (3), (5), (6) in the ring ℤ/14ℤ. There are :

• a) Only one ideal in this list.
• b) Two distinct ideals in this list.
• c) Three distinct ideals in this list.
• d) Four distinct ideals in this list.

I know the answer is related to divisors of 14, but I am not sure which answer is correct. I would eliminate a) and d), and my feeling is that the right answer would be c).

Thanks.

Answer 1

While you can explicitly enumerate the elements of each ideal and compare them (as suggested by @jammarqz), I like the following approach:

Note that there is a natural isomorphism $(\mathbb{Z}/(14))/(n)\cong \mathbb Z / (14,n)$, but $\mathbb Z$ is a PID and the ideal generated by $14$ and $n$ will be the ideal generated by their GCD. So all you have to do is compute the GCD of $14$ and the numbers in the question.

Answer 2

Hint. The multiples $(2)=\{2,4,6,8,12,0\}$ is an additive subgroup, but also ideal.

But also note that $(3)=\{3,6,9,12,1,4,7,10,13,2,5,8,11,0\}$, which is the whole ring.

Answer 3

The ideal $(n)$ in $\mathbb Z/14\mathbb Z$ is the image of the ideal $(n,14)$ in $\mathbb Z$ under the canonical quotient map.

Now $(n,14)$ in $\mathbb Z$ is equal to $\gcd(n,14) \mathbb Z$.