The sentence you are citing, "For each $i$, there exists $x_i\in I$ such that $x_i\notin p_k$ for all $k\neq i$", is not the logical negation of the statement, but the application of induction. That is, they assume that the statement is true for $n-1$ prime ideals and are applying to all subsets of size $n-1$ of $\{p_1,\ldots, p_n\}$, i.e., the set of the form $\{p_j\,|\,j\neq i\}$ with $i$between $1$ and $n$

Logic problem: Atiyah-Macdonald 1.11
I read the following question: Tips for a first year CS student looking for a summer internship to gain experience?
But rather than how to get and/or look for internships, my questions concerns of how to filter companies looking for free labor to do their website VS companies that might not offer a paid check, but it will offer you mentorship and skills.

I have a cousin that is a college freshman and she is looking for Sofware development internships, but at this point, she is desperately looking for any internship that is IT related. Besides the big software names and elite small software shops, How do you recognize a good internships in non-it-companies(large or small) doing IT maintenance vs the bad ones? If you don't have work experience, what questions a college student should ask to prevent ripoff? Are there any red flags that you can spot before accepting an offer?
