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Assuming we have a ring which is probably not a unique factorization domain, for example $$F_{p}[x]/(x^{4}-x^{2})$$

Is there a clean way to detect which elements in this ring are prime? We can write down the irreducible elements very easily by the degree; but how do we know which one of them is prime?

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I think you are mixing two concepts. It can happen that there is no unique factorization for elements in a ring while there is unique factorization for ideals --- this happens in the ring of algebraic integers in a number field. In such a setting, irreducible elements may not be prime elements, but irreducible ideals are prime ideals. –  Gerry Myerson Nov 13 '12 at 23:20
That ring is not a UFD since it is not even a domain. Factorization theory is more complicated in non-domains. Basic notions such as associate and irreducible bifurcate into a few inequivalent notions, e.g. see here. As it stands, the question is too general. It cannot be reasonably answered until it is localized to some class of rings with nice factorization properties. –  Bill Dubuque Nov 13 '12 at 23:35
To elaborate a little on Gerry Myerson's comment, such domains are in general called Dedekind domains. I'm not sure how, if at all, it generalizes to non-domains. –  tomasz Nov 13 '12 at 23:38
I see. This is a bad confusion. I should clarify. –  Bombyx mori Nov 14 '12 at 1:19

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