Consider the ideal I=<a,b> generated by a,b in R. As R is a PID, I=<a,b>=<c> for some "c\\in R."
This imply c|a, c|b, which imply c is a unit as a,b are coprime.
Therefore 1"\\in <a,b> \\Rightarrow 1=ax+by" for some x,y"\\in R" .
Converse
let c|a, c|b. Then c|ax+by "\\Rightarrow c|1".
Therefore c is a unit, which helps us to conclude that a,b are coprime.
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