Notation Question about Rings

Question

If $S = \langle2\rangle$ is the ideal generated by $2$ in $\mathbb{Z}$, what does $S[x]$ represent?

Answer 1

$S[x]$ is the ideal of polynomials with even coefficients. Generally if $I \subset R$ is an ideal and $M$ is a module, the $IM$ is the submodule of $M$ generated by elements of the form $im$, where $i \in I$ and $m \in M$. Sometimes you can use shortcuts, for example $I R[x] =: I[x]$.

It's not maximal in $\mathbb{Z}[x]$: the quotient $\mathbb{Z}[x] / S[x]$ is equal to $(\mathbb{Z}/2\mathbb{Z})[x]$, which is not a field. Another way of seeing it is to notice that $S[x]$ is included in the ideal generated by $2$ and $x$ -- in other words the ideal of polynomials with even constant term.