$\langle 2,X\rangle$ in the polynomial ring $\mathbb{Z}[X]$

Question

Why is that $\langle 2,x\rangle$ in $\mathbb{Z}[X]$ represent the set: $\{2g(x)+xf(x)\}$?

I'm just a little unclear about the notation. What element does this generator exactly produce in this ring?

Answer 1

$\langle 2,X\rangle$ is by definition the smallest ideal of $\Bbb Z[X]$ that contains the listed generators $2$ and $X$. Being an ideal, $\langle 2,X\rangle$ thus certainly contains anything that can be written as $2g(X)+Xf(X)$. On the other hand, one readily verifies that the set $\{\,2g+Xf\mid g,f\in\Bbb Z[X]\,\}$ is an ideal, hence we have equality.

Answer 2

$\langle 2, x \rangle$ denotes the smallest ideal of $\mathbb{Z}[x]$ containing the elements $2$ and $x$.

Exercise: Prove that the intersection of all ideals containing $2$ and $x$ is an ideal containing $2$ and $x$. This justifies the claim that there is a smallest such ideal.

Ideals are closed under addition, and closed under multiplication by ring elements. Every element of $\mathbb{Z}[x]$ of the form $\alpha \cdot 2 + \beta \cdot x$ (where $\alpha, \beta \in \mathbb{Z}[x]$) can be obtained by applying those operations to $2$ and $x$.

Every ideal containing $2$ and $x$ must therefore contain every element of the form $\alpha \cdot 2 + \beta \cdot x$. Since that set is an ideal, we infer it is the (set of elements of) $\langle 2 , x \rangle$.

Exercise: Show that this set is, indeed, the set of elements of an ideal of $\mathbb{Z}[x]$.