Question

$$\begin{gathered} \mathit{P}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathbf{}\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mathbf{}\mathit{e}\mathit{v}\mathit{e}\mathit{r}\mathit{y}\mathbf{}\mathit{i}\mathit{d}\mathit{e}\mathit{a}\mathit{l}\mathbf{}\mathit{i}\mathit{n}\mathbf{}\mathit{Z}\mathbf{}\mathit{i}\mathit{s}\mathbf{}\mathit{p}\mathit{r}\mathit{i}\mathit{n}\mathit{c}\mathit{i}\mathit{p}\mathit{a}\mathit{l}\mathbf{.}\mathbf{}\mathbf{[}\mathit{H}\mathit{i}\mathit{n}\mathit{t}\mathbf{:}\mathbf{}\mathit{I}\mathit{f}\mathbf{}\mathbf{}\mathit{i}\mathit{s}\mathbf{}\mathit{a}\mathbf{}\mathit{n}\mathit{o}\mathit{n}\mathit{z}\mathit{e}\mathit{r}\mathit{o}\mathbf{}\mathit{i}\mathit{d}\mathit{e}\mathit{a}\mathit{l}\mathbf{,}\mathbf{} \\ \mathit{s}\mathit{h}\mathit{o}\mathit{w}\mathbf{}\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mathbf{}\mathit{I}\mathbf{}\mathit{m}\mathit{u}\mathit{s}\mathit{t}\mathbf{}\mathit{c}\mathit{o}\mathit{n}\mathit{t}\mathit{a}\mathit{i}\mathit{n}\mathbf{}\mathit{p}\mathit{o}\mathit{s}\mathit{i}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{}\mathit{e}\mathit{l}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{s}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{,}\mathbf{}\mathit{h}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{,}\mathbf{}\mathit{m}\mathit{u}\mathit{s}\mathit{t}\mathbf{} \\ \mathit{c}\mathit{o}\mathit{n}\mathit{t}\mathit{a}\mathit{i}\mathit{n}\mathbf{}\mathit{a}\mathbf{}\mathit{s}\mathit{m}\mathit{a}\mathit{l}\mathit{l}\mathit{e}\mathit{s}\mathit{t}\mathbf{}\mathit{p}\mathit{o}\mathit{s}\mathit{i}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{}\mathit{e}\mathit{l}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{c}\mathbf{}\mathbf{}\mathbf{(}\mathit{W}\mathit{h}\mathit{y}\mathbf{?}\mathbf{)}\mathbf{.}\mathbf{}\mathit{S}\mathit{i}\mathit{n}\mathit{c}\mathit{e}\mathbf{}\mathit{c}\mathbf{\in }\mathit{I}\mathbf{}\mathit{e}\mathit{v}\mathit{e}\mathit{r}\mathit{y}\mathbf{}\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i}\mathit{p}\mathit{l}\mathit{e}\mathbf{} \\ \mathit{o}\mathit{f}\mathbf{}\mathit{c}\mathbf{}\mathit{i}\mathit{s}\mathbf{}\mathit{a}\mathit{l}\mathit{s}\mathit{o}\mathbf{}\mathit{i}\mathit{n}\mathbf{}\mathit{I}\mathbf{}\mathbf{;}\mathbf{}\mathit{h}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{,}\mathbf{}\mathbf{\left(}\mathbf{c}\mathbf{\right)}\underline{\mathbf{\subset }}\mathit{I}\mathbf{}\mathit{T}\mathit{o}\mathbf{}\mathit{s}\mathit{h}\mathit{o}\mathit{w}\mathbf{}\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mathbf{}\mathit{I}\underline{\mathbf{\subset }}\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{,}\mathbf{}\mathit{l}\mathit{e}\mathit{t}\mathbf{}\mathit{b}\mathit{e}\mathbf{}\mathit{a}\mathit{n}\mathit{y}\mathbf{}\mathit{e}\mathit{l}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{I} \end{gathered}$$$\mathit{T}\mathit{h}\mathit{e}\mathit{n}\mathbf{}\mathit{a}\mathbf{=}\mathit{c}\mathit{q}\mathbf{+}\mathit{r}\mathbf{}\mathit{w}\mathit{i}\mathit{t}\mathit{h}\mathbf{}\mathbf{0}\mathbf{\le }\mathit{r}\mathbf{<}\mathit{c}\mathbf{}\mathbf{(}\mathit{W}\mathit{h}\mathit{y}\mathbf{?}\mathbf{)}\mathbf{.}\mathbf{}\mathit{S}\mathit{h}\mathit{o}\mathit{w}\mathbf{}\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mathbf{}\mathit{r}\mathbf{=}\mathbf{0}\mathbf{}\mathit{s}\mathit{o}\mathbf{}\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mathbf{}\mathit{a}\mathbf{=}\mathit{c}\mathit{q}\mathbf{\in }\left(c\right)\mathbf{.}\mathbf{]}$

Short answer

Every Z in ideal is principal.

Step-by-step solution

Write the division algorithm theorem

Consider $a,b$be integers with $b>0$. Then, there exists unique integrs $q$and $r$such that $a=bq+r$ and $0\le r<b$.

Show that every ideal in Z is principal

Assume$I$ be a non-zero ideal$\cancel{c}$ in and assume$c\in I$ as the smallest positive integer contained in $I$.

As every multiple of$c$ is in$I$ then, we have $\left(c\right)\underline{\subset }I$. Assume $a\in I$.

Apply division algorithm. Then, we have,

$q,r\in \cancel{c}$such that$a=qc+r$ and $0\le r<c$.

Since$I$ be a non-zero ideal then, $a-qc=r\in I$by the definition of division algorithm$c$ is the smallest positive integer in $I$then we get, $r=0$so that $a=qc$and $a\in \left(c\right)$.

Since$a\in I$ was chosen arbitrarily, the n $I\underline{\subset }\left(c\right)$and thus, $I=\left(c\right)$.

Hence, the given statement is Proved.