Answer to Question #139110 in Abstract Algebra for anjali

Question #139110
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
1
Expert's answer
2020-10-20T17:41:14-0400

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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