I am confused as to how to deal with basic polynomials in the sense of principal ideals.
For example, let $I=\lbrace p(x) \in \mathbb{Z}[X] : p(0)=7j \text{ for some } j\in \mathbb{Z} \rbrace$ be an ideal of $\mathbb{Z}[X]$ (the ring of all polynomials with integer coefficients). I am wanting to show that $I$ is not a principal ideal of $\mathbb{Z}[X]$. We know that $x+7\in I$ and $x+14\in I$, and also that these elements are not multiples of each other. What does not being multiples of each other mean in this context? In other words, how does this show that $I$ is not principal ideal of $\mathbb{Z}[X]$? Any help would be appreciated.