The original answer here was incorrect and I removed it. There is a different answer "usingAnother example using idealization construction" submitted with this questionconstruction (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 $K[X_1,X_2,...]$