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Lemma: Greatest Common Divisor and Least Common Multiple of Ideals

Let \(I\lhd R\) and \(J\lhd R\) be two ideals of the ring \(R\). Then

The sum \(I+J\) is an ideal of \(R\) and is the greatest ideal dividing \(I\) and \(J\). We denote it as the greatest common divisor of the ideals

\[I+J=gcd(I,J).\]

The intersection of the ideals \(I\cap J\) is an ideal of \(R\) and is the smallest ideal, which is divisible by \(I\) and \(J\). We denote it as the least common multiple of the ideals

\[I\cap J=lcm(I,J).\]

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Proofs: 1

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References

Bibliography

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