What would an ideal I generated by e.g. 15 and 12 look like? What would the quotient ring $\mathbb{Z} / I$ look like?

Question

What would an ideal I generated by e.g. 15 and 12 look like?

What would the quotient ring $\mathbb{Z} / I$ look like? How do I find a formal representation for this quotient ring?

Thanks

Answer 1

In general, if $I=(n,m)$ then $I=(gdc(n,m))$. Therefore $I=(15,12)=(3)$ and the quotient is the field $\mathbb{Z}/(3)=\mathbb{Z}_{3}$.

Answer 2

It is the ideal generated by $\gcd(12,15)$, hence the quotient is $$\mathbf Z/I=\mathbf Z/3\mathbf Z.$$

Answer 3

Let's see why the ideal is indeed generated by the gcd.

Recall the definition of a generated ideal. In this case:

$$I = \{ 15m + 12n = 3(5m + 4n) : m,n \in \mathbb {Z}\}$$

Now, by Bezout/Reverse Euclid, since $5$ and $4$ are coprime, we can find $p, q $ st $5p + 4q = 1$ (in this case $p=1, q=-1$ will do) and hence $5pr+4qr = r $ for any $r \in \mathbb {Z} $.

Hence everything of the form $3r $ is in $I$, and you can ses that there's nothing else in the ideal. Therefore, $I$ is the ideal generated by $3$.

As for the quotient, one useful way to think about them is that $\text {"R/I is R, but with elements of I treated as 0"}$. In this case, the quotient is $\text {"$\mathbb {Z}$, but with multiples of 3 treated as $0$}"$. This should sound familiar!