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I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them there are idealizations and valuation domains. But the first non-noetherian ring we are thinking about is $K[X_1,\dots,X_n,\dots]$, $K$ a field. The finitely generated ideals of this ring have primary decomposition, so if they are irreducible then are necessarily primary.

My question is the following:

Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?

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Another example using idealization construction (i.e. $(a,b)(c,d)=(ac,ad+bc)$):

  1. start with $R=k[T]_{(T)}+k(T)$, with $k=$prime field contained in $K$;
  2. primary ideals of $R$ are $(0)+(0)$, $(0)+k(T)$, and $(T^n)+k(T)$ $(n>0)$;
  3. the ideal $I=(0)+k[T]_{(T)}$ is irreducible and not primary;
  4. $R$ is countable so it's a quotient of $k[X_1,X_2,...]$;
  5. extend scalars from $k[X_1,X_2,...]$ to $K[X_1,X_2,...]$
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  • $\begingroup$ This looks fine. The only step in doubt is (5) since it's not so clear (to me) that (non)primary, respectively irreducible ideals behaves well when extending in faithfully flat extensions. $\endgroup$
    – user26857
    Commented Nov 13, 2014 at 22:42
  • $\begingroup$ They behave well in this example. Extension of scalars gives a 1-1 correspondence between the ideal lattices i.e. $\endgroup$
    – David Lampert
    Commented Nov 14, 2014 at 0:21
  • $\begingroup$ $(0)+(0),...,(0)+(T^3),(0)+(T^2),(0)+(T),(0)+(1),\ldots,(T^3)+(1),(T^2)+(1),(T)+(1),(1)+(1)$. (Sorry, I'm not fluent with math markup here.) $\endgroup$
    – David Lampert
    Commented Nov 14, 2014 at 0:38

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