I can give [examples](http://math.stackexchange.com/a/965703/121097) of non-noetherian rings having irreducible ideals that are not primary. Among them 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 $K[X_1,\dots,X_n,\dots]$?