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:



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> Are there irreducible ideals that are not primary in $K[X\_1,\dots,X\_n,\dots]$?
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Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?
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 $K[X\_1,X\_2,...]$


