TITLE: Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?
QUESTION [11 upvotes]: 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 $K[X_1,\dots,X_n,\dots]$?

REPLY [2 votes]: Another example using idealization construction (i.e. $(a,b)(c,d)=(ac,ad+bc)$):

start with $R=k[T]_{(T)}+k(T)$, with $k=$prime field contained in $K$;
primary ideals of $R$ are $(0)+(0)$, $(0)+k(T)$, and $(T^n)+k(T)$ $(n>0)$;
the ideal $I=(0)+k[T]_{(T)}$ is irreducible and not primary; 
$R$ is countable so it's a quotient of $k[X_1,X_2,...]$;
extend scalars from $k[X_1,X_2,...]$ to $K[X_1,X_2,...]$