Question

In the Principal Ideal Domain two non-zero element a,b are coprime if and only if ∃ x,y∈ R such that ax+by=1

Answer 1

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.