Question

Let $\mathit{J}$be the set of all polynomials with zero constant term in $\mathit{\mathbb{Z}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$.

(a) Show that $\mathit{J}$is the principal ideal $\mathbf{\left(}\mathbf{x}\mathbf{\right)}$in$\mathit{\mathbb{Z}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$ .

Short answer

It is shown that$J=\left(x\right)$

Step-by-step solution

Consider the set 

Consider that $x\in J$, and the function$f\left(x\right)\in J$.

Show that J  is the principal ideal (x) in ℤ[x]

Write the function in the form as shown below:

$f\left(x\right)=\sum _{i=1}^k{a}_i{x}^i=\left(\sum _{i=1}^k{a}_i{x}^{i-1}\right)x$

Thus, $f\left(x\right)\in \left(x\right)$,and $J\subseteq \left(x\right)$.

Hence, it is concluded$J=\left(x\right)$.