Let $(2)\in Z[x]$ be the Z[x]-ideal generated by 2. Is there any difference in elements of (2) if we view this ideal as $Z[x]-module$ compare to an $Z-module$?
Because, in my attempt, by definition we have $(2)=\{p(x)*2 | p(x) \in Z[x]\}$, But on the other side, in a $Z-module$ the multiplication defined as $z*r$ (when $z \in Z $ and $r \in R$), so perhaps when viewed as $Z-module$ the ideal is $(2)=\{z*2 | z \in Z\}$?
Thanks!
Well, for instance $2x \in (2)$ as a $\mathbb{Z}[x]$-module while $2x \notin (2)$ as a $\mathbb{Z}$-module. As someone mentioned in the comments, as a $\mathbb{Z}[x]$-module we have $$(2) = \{ 2p(x) \mid p(x) \in \mathbb{Z}[x] \}$$ while as a $\mathbb{Z}$-module we have $$(2) = \{ 2k \mid k \in \mathbb{Z}\}.$$