gcd & lcm in a PID

Question

In a PID, $l={\rm lcm}(a,b)$ and $d=\gcd(a,b)$. Is it always true that the following product ideals are equal? $$<d><l> = <a><b>$$

Thanks in advance

-- Mike

Answer 1

A PID is a UFD. Let $$a=p_1^{r_1}p_2^{r_2}\cdots p_t^{r_t},\quad b=p_1^{s_1}p_2^{s_2}\cdots p_t^{s_t}$$ for primes $p_1,p_2,\dots,p_t$. Then $$\ell={\rm lcm}(a,b)=\prod p_j^{\max(r_j,s_j)},\quad d=\gcd(a,b)=\prod p_j^{\min(r_j,s_j)}$$ so $$\ell d=\prod p_j^{\max(r_j,s_j)+\min(r_j,s_j)}=\prod p_j^{r_j+s_j}=ab$$