Let $R$ be an integral domain. Now suppose every finitely generated ideal of $R$ is principal; I conjecture that for any $a, b \in R$, at least one of them is invertible.
Is this a valid conjecture ? If yes, how do I make use of the principality property?
The reason behind the conjecture is that I have been tasked to show that the principality property stated can in fact be used as a definition for valuation ring.
Any help or insights is appreciated
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