Let \((R, + , \cdot)\) be an integral domain and let \((i)\lhd R\), \((j)\lhd R\) be two principal ideals, i.e. there exist two elements \(i,j\in R\) with \((i)=Ri\) and \((j)=Rj\). In this case, the following equivalence holds:
$$i \mid j \Longleftrightarrow (i) \mid (j),$$
or, in set-theoretic notation,
$$i \mid j \Longleftrightarrow (i)\supseteq (j).$$
Proofs: 1
