Why the ideals here are in this form?

Question

In this article, in the proof of problem no. 6, p. 3, it listed all the possible ideals, because they contains at least one of $2$, $3$, $5$, from $120$. And at least one of $x+1$, $x^2-x+1$. But I think it is also possible to contain $2$ elements from $2$, $3$, $5$. And $2$ elements from $x+1$, $x^2-x+1$. In other words, why it is not possible for the ideal of the form $(2, 3, x+1)$? In addition, even if in the case it contains only $2$ and $x+1$,does it mean the ideal should be generated by $2$ and $x+1$,i.e., the ideal $(2, 3, x+1)$? Apparently if an ideal contains $2$ and $x+1$, it can be possible in the form like $(2, 7, x+1)$. I feel confused about this one. Could you give me some ideas? Thank you!

Answer 1

Once an ideal in $\Bbb{Z}[x]$ contains both $2$ and $3$, it contains $\operatorname{gcd}(2, 3) = 1$. Then the ideal is all of $\Bbb{Z}[x]$. It's not that you can't have such an ideal; it's just redundant. You can describe it with fewer generators (and you should).