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The point of UFD is that element can be factored finitely and uniquely. One non-example is $Z[\sqrt{-5}]$ where there may be non-unique factorization. I wonder if there is any simple example that violates the finiteness requirement, i.e. there is some element that can be factored again and again into non-unit elements.

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marked as duplicate by Bill Dubuque abstract-algebra Dec 5 '18 at 19:33

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In the ring of all algebraic integers, every nonzero nonunit is a square, for example. In this case, there aren't any irreducible elements at all.

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