In $\mathbb Z[x]$, is $(2,x)=(2)+(x)$?

Question

The text says that $(2,x)=(2)+(x)$, because $1 \in \mathbb Z$. I do not see why this leads to the decomposition.

Can someone point me in the right direction?

Answer 1

Let $R$ be a ring and $a,b\in R$. Then $(a)$ is the smallest ideal containig $a$, $(b)$ is the smallest ideal containing $b$, and $(a,b)$ is the smallest ideal containing both $a$ and $b$. The sum of ideals $(a)+(b)$ is an ideal and clearly contains $a=a+0$ and $b=0+b$, hence $(a,b)\subseteq (a)+(b)$. On the other hand, as $(a)\subseteq (a,b)$ and $(b)\subseteq (a,b)$, also $(a)+(b)\subseteq (a,b)$. We do not need a unit to conclude $(a,b)=(a)+(b)$.

However, without a unit it may happen that $(a)\ne aR$.

Answer 2

$(2,x)=\left\{f\cdot2+g\cdot x|f,g\in\mathbb{Z}[x]\right\}=\left\{f+g|f\in(2),g\in(x)\right\}=(2)+(x)$