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Exhibit an integral domain $R$ and a non-zero non-unit element of $R$ that is not a product of irreducibles.

My thoughts so far: I don't really have a clue. Could anyone direct me on how to think about this? I'm struggling to get my head round irreducibles.

Thanks.

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    Hint: if $R$ is noetherian, any element is a product of irreducible ones. Do you know any non-noetherian ring?2011-05-06
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    wouldn't all fields work for this??2011-05-06
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    @quanta: In a field, there are no nonzero nonunits. If the statement was "an integral domain R such that for all nonzero nonunit elements a, a is not a product of irreducibles", then a field wouldd satisfy it by vacuity. But here they are asking to *exhibit* an $R$ and an $a$, so fields don't work.2011-05-07

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