Exercise 2.2.14 (Ideal is equal to principal ideal)

Question

Fill in the details of Example 2.2.13.

Hassaan Naeem · Accepted Answer

\paragraph{Solution:} We suppose that $I \subseteq \mathbb{Z}$ is an ideal and we take $n \in I$ to 
be the least positive integer in $I$. We have obviously that $\langle n angle \subseteq I$. Then we assume that
that $m \in I$, by the division algorithm we know that:

\begin{equation*}
        \begin{aligned}
            m &= qn +r  \quad &(0 \leq r < n)\
            r &= m - qn \quad &\in I
        \end{aligned}
    \end{equation*}
Therefore $r=0$ $\leadsto m = qn$. Therefore $m \in \langle n angle$ and we have that $I \subseteq \langle n angle$.
Hence we have equality, $I = \langle n angle \quad \square$

Exercise 2.2.14 (Ideal is equal to principal ideal)

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Exercise 2.2.14 (Ideal is equal to principal ideal)

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