Question

Prove (1) in Theorem G.4

Short answer

1) in Lemma G.4 is proved.

Step-by-step solution

Introduction of polynomial

Consider the polynomial in $\mathbf{x}$ and $\mathbf{ℝ}$will be express as;

$\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{a}}_{\mathbf{0}}{\mathbf{x}}^{\mathbf{0}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{x}}^{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{2}}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{...}\mathbf{+}{\mathbf{a}}_{\mathbf{n}}{\mathbf{x}}^{\mathbf{n}}$

Polynomial is always finite but sequence can be finite and infinite.

Prove (1) in Lemma G.4

Consider the theorem G.4.

Consider$P$ be the ring with coefficient in the ring$R$ and$P$ contains an isomorphic copy $R^{*}$of$R^{}$ and an element $x$androle="math" localid="1658912761906" $a\in R^{*}$ , and$a=(a,0_R,0_R,0_R,\mathrm{...})$

Then,

Hence,$ax=xa$

Therefore, (1) in LemmaG.4 is proved.