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I am looking for an example of a primary decomposition of an ideal in a commutative noetherian ring, that isn't an UFD nor a dedekind domain and where the primary ideals in the decomposition is not equal to there radical ideals. That is, a good example on how to use the theory of existence and uniqueness of primary decomposition in Noetherian rings.

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-navigetor23. The examples given in Atiyah and MacDonald (section 4 about Primary decomposition) concerns the ring $K[x_1,x_2,...,x_n]$, for $K$ a field and various $n>0$. This is a UFD, so no, there is no such examples given there. –  harajm Nov 19 '12 at 18:51
    
@harajm, some examples are about quotients of polynomial rings, i.e. it's not UFD/Dedekind at all. –  user27126 Nov 19 '12 at 19:33
    
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