Proof of a coprimes property in a UFD

Question

I don't know how to prove well that if D is an UFD and $a,b\in D$ coprimes then if $a\mid bc\Rightarrow a\mid c$. I've tried to put the definitions but I don't know how to conclude that $a\mid c$

Answer 1

Hint:

The uniqueness, up to a unit factor, of factorisation of elements of $D$ into prime factors shows a prime factorisation of $bc$ is obtained multiplying a prime factorisation of $b$ and a prime factorisation of $c$. You can proceed by induction on the number $r$ of prime factors of $a$:

If $r=1$, the ideal $(a)$ is prime and the hypothesis says $bc\in(a)$, but $b\notin (a)$, so by definition, $c\in (a)$.

Induction step: if $r>1$, write $a=pa'$, where $p$ is one of the prime factors of $a$, and $a'$ has $r-1$ prime. factors. Note that no prime factor of $a$ (hence of $a'$) divides $b$. Can you proceed?