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Suppose $A$ is a principal ideal domain with every ideal of finite index (except the zero ideal). Must $A$ be a Euclidean domain?

If it's not known, are there any relevant partial results?

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Not necessarily. The classical example of PID which is not Euclidean, $R=\mathbb Z[\frac{1+\sqrt{-19}}{2}]$ has the property that $R/I$ is finite for all $I\ne(0)$. (In order to prove this use a similar argument to the one given in this answer.)

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  • $\begingroup$ Exactly what i was looking for, thanks. $\endgroup$ – Vik78 Feb 3 '16 at 17:05
  • $\begingroup$ @Vik78 Welcome. $\endgroup$ – user310443 Feb 3 '16 at 17:06

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