I was thinking about how a chain of irreducible polynomials in $K[X_1,\ldots,X_n]$, where $K$ is a field, behave with respect of being prime. What I mean is the folowing:
If $\{f_1,\ldots,f_n\}$ is a sequence of irreducible polynomials in $K[X_1,\ldots,X_n]$, I know that the ideal generated by $(f_i)$ is prime for every $i = 1,\ldots,n$. But what I can have from the ideals of the form $I_k = (f_1,f_2,\ldots,f_k)$? Are these ideals $I_k$ prime?
I'm asking this because if this is true there is a very simple answer for my question here.
Thank you for the help!!