1
$\begingroup$

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?

$\endgroup$
1
  • 1
    $\begingroup$ I like it, it's correct. $\endgroup$ – Mars Nov 19 '20 at 5:14
1
$\begingroup$

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 $b$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.