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 ?
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user26857
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No, e.g. the ring of all algebraic integers or entire functions. – Bill Dubuque Mar 26 '15 at 13:57
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@BillDubuque : Thanks , can you please provide an answer proving the set of all algebraic integers satisfies the condition but is not a PID ? (I don't know about Entire functions ) are there any other simpler examples ? – Mar 26 '15 at 14:00
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See the link in this answer. – Bill Dubuque Mar 26 '15 at 14:03
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http://en.wikipedia.org/wiki/B%C3%A9zout_domain – Qiaochu Yuan Mar 26 '15 at 16:56
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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.
Bill Dubuque
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