In a principal ideal domain, gcd(a,b) always exists and can always be expressed as $xa +yb$ with some $x, y in R$.
I am quite new to algebra and would be really grateful if someone could show me ow to prove this.
2026-03-25 14:36:38.1774449398
In a principal ideal domain, gcd(a,b) always exists and can always be expressed as $xa +yb$ with some $x, y /in R$
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Consider the ideal $I := \{ xa + yb \mid x, y \in R \}$ (why is this an ideal?).
Since it is principal, it is generated by some element, say $r \in I$.
Show that $r$ is a gcd of $a$ and $b$.