Least common multiples exist in a PID

Let R be a principal ideal domain. Prove that any two nonzero elements in R have a least common multiple.

We saw in this previous exercise that the least common multiples of a and b are precisely the generators of the \subseteq-largest principal ideals contained in (a) \cap (b). Since R is a principal ideal domain, (a) \cap (b) = (\ell) is itself principal, so that a least common multiple indeed exists.

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  • DL  On November 28, 2011 at 9:28 pm

    There is some typo,,, 🙂

    • nbloomf  On November 29, 2011 at 12:05 am


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