Question

Check whether or not S = { a0 + a1x + .... + an(x^n) belongs to Z[x] | 5 | a0} is an ideal of Z[x].

Answer 1

Let $f\in\mathbb Z[x],\,\,f=b_0+b_1x+\dots$ and $s\in S$

$fs=a_0b_0+\dots \in \mathbb Z[x]$ and $5\mid a_0'=a_0b_0$

Hence, $fs\in S$ and $S$ is an ideal of $\mathbb Z[x]$