(related to Lemma: Divisibility of Principal Ideals)

- By assumption, \((i)=Ri\) and \((j)=Rj\) are principal ideals for some \(i,j\in R\), where $(R, + ,\cdot)$ is an integral domain.
- If \(i \mid j\), then following the definition of divisibility in the ring there exists an element \(x\in R\) with \(xi=j\).
- It follows that $(j)=Rj=R(xi)\subseteq Ri=(i).$
- Following the definition of divisibility of ideals $(j)\subseteq (i)$ is equivalent to $(i)\mid (j).$

- If, vice versa, \((i)\mid (j)\), we have the set inclusion \((j)\subseteq (i)\).
- Because \(j\in (j)\), it follows that \(j\in (i)\).
- Therefore, there exists an element \(x\in R\) with \(j=xi\).
- Thus, \(i\mid j\).∎

**Kramer Jürg, von Pippich, Anna-Maria**: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013