# In a PID $R$, given $a,b\in R$, what are the ideals $\langle a\rangle,\langle b\rangle$ also generated by?

Let $$R$$ be a principal ideal domain and let $$a$$ and $$b$$ be two nonunit elements in $$R$$ then ideal generated by $$a$$ and $$b$$ is also generated by

1. $$a+b$$

2. $$ab$$

3. $$\gcd(a,b)$$

4. $$\text{lcm}(a,b)$$

I am not sure but I think that answer is 3.

Let $$I=\langle a,b \rangle$$ $$=\{ax+by:x,y\in R\}$$. Let $$d=gcd(a,b)$$. Since $$R$$ is a PID we get $$d=ax+by$$ for some $$x,y\in R$$. So $$\langle d \rangle \subseteq I$$ as $$d$$ is a divisor of $$a$$ and $$b$$ clearly $$a,b \in \langle d \rangle$$ So $$I \subseteq \langle d \rangle$$. Hence $$I=\langle d \rangle$$. Is it correct?

• I like it, it's correct. – Mars Nov 19 '20 at 5:14

Yes, it is correct. In fact, some authors go a step further and define the gcd of $$a$$ and $$b$$ of a ring to be a generator of the smallest principal ideal containing both $$a$$ and $$b$$.