# Thread: factorial domains

1. ## factorial domains

Any thoughts?

Suppose R is a factorial domain and a and b are in R with a not zero and a and b relatively prime. Show that if a divides bc, c in R, then a divides c.

Thanks!

2. Hi

Lemma (Maybe from Gauss): Let $\displaystyle p$ be an irreducible element in a UFD $\displaystyle R,$ then for any elements $\displaystyle a,b$ in $\displaystyle R,$ $\displaystyle p|ab\Rightarrow p|a\ \text{or}\ p|b$

In your case, any element can be written as a (unique) product of irreducible elements.

a and b relatively prime
Write what this means, and the solution should come.

EDIT: As ThePerfectHacker said, what I originally wrote was wrong.

3. Originally Posted by clic-clac
Lemma (Maybe from Gauss): Let $\displaystyle p$ be an irreducible element in a ring $\displaystyle R,$ then for any elements $\displaystyle a,b$ in $\displaystyle R,$ $\displaystyle p|ab\Rightarrow p|a\ \text{or}\ p|b$
This is not true. If R is a UFD then it is, but not in general.