Lemma: Divisibility of Principal Ideals

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

Proofs: 1 2


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References

Bibliography

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