Let $D$ be an integral domain such that for any $a,b \in D$, $Da+Db$ is a principal ideal. Then must $D$ necessarily be a principal ideal domain i.e. should all the ideals of $D$ be principal ?
2026-03-27 02:35:31.1774578931
Does validity of Bezout identity in integral domain implies the domain is PID?
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Common examples of non-PID Bezout domains are the rings of all algebraic integer or entire functions, e.g. this answer. For a simpler example one may consider the semigroup ring $\, F[x^{\Bbb Q_{\ge 0}}].\, $ Below is a sketch of this example from M. S. Osborne's Basic Homological Algebra, p. 92.