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Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$?

It would be preferable if examples are something that does not involve monomials/polynomials, and there are some integer parts in the non-UFD (for example, gaussian integers have integer parts, though gaussian integer is UFD).

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Let $\,a = 2,\ b=\sqrt{12}\,$ in $\,\Bbb Z[\sqrt{12}]$

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  • $\begingroup$ +1 a non-maximal order is surely the simplest way to go. At least if we insist that the ring should be a domain (without that requirement the question is rather dull). $\endgroup$ – Jyrki Lahtonen Aug 20 '14 at 6:16
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Let $a=4$, $b=8$ in $4{\bf Z}$.

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  • $\begingroup$ That's as may be, but OP specifically asked for a non-UFD. $\endgroup$ – Gerry Myerson Aug 20 '14 at 23:02

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