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principal ideal domain

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Let R be a principal ideal domain and a, b, in R, not both zero. Prove that a, b have a greatest common divisor that can be written as linear combination of a and b. Hint: let I be the ideal generated by a and b, then I = (d) for some d in R. Show that d is a gcd of a and b.

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Solution Summary

This solution clearly assesses the principal ideal domain in this case.

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Proof:
We consider the ideal I = <a,b> generated by a and b. Since R is a principal ideal domain, then we can find
some d in R, such that I = <d> = <a,b>. Then a is in I = <d> and ...

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