I'm just getting started learning ring theory and am currently learning about ideals. By book (Dummit & Foote) says the following:
For example, in the ring $R = \mathbb{Z}[x]$ the elements $2$ and $x$ generate a maximal, non-principal ideal
First off, for clarification, the notation $\mathbb{Z}[x]$ just means "the set of all polynomials with integer coefficients," right (i.e. the set of all
$$ a_0 + a_1x + a_2x^2 + \cdots $$
such that each $a_i \in \mathbb{Z}$)? Second, how is the ideal generated by $2$ and $x$ maximal? If you let $I$ be the ideal generated by $2$ and $x$ and let $J$ be the ideal generated by $1$ and $x$, isn't it true that $I \subset J$ and so $I$ is not maximal?
Any clarification would be greatly appreciated!