Let $R$ be a Factorization domain (https://en.m.wikipedia.org/wiki/Atomic_domain ) which is local and with principal maximal ideal. Then is $R$ a Bezout domain i.e. every finitely generated ideal is principal ? Or equivalently (since $R$ is local), is $R$ a valuation domain ( https://en.m.wikipedia.org/wiki/Valuation_ring) i.e. given any non-zero $a,b \in R$, either $a|b$ or $b|a$ ? Since factorization domain which are Bezout are exactly PID, and local PID is a DVR, so equivalently we ask : is $R$ a DVR ?
1 Answer
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Suppose $R$ isn't a field. If $M$ is the maximal ideal and $M=(p)$, then $p$ is the only irreducible element. For suppose $q$ was another irreducible: then $q=pr$ implies $r$ is a unit, so that $p$ is associate to $q$.
Moreover, $p$ is prime since $R/M$ is a field. Thus you are talking about an atomic domain in which all irreducibles are prime, a.k.a. a UFD. (You probably don't even need this step but it's worth mentioning.)
Everything uniquely factors into the form $p^nu$ where $n\in\mathbb N$ and $u$ is a unit. This implies the principal ideals are linearly ordered, and it is easy to show this implies all ideals are linearly ordered.
So yes, it is a valuation ring.
