# How to prove that two ideals (a) and (b) in PID are comaximal iff gcd(a,b) = 1

The best way to tackle this problem is by using the definition of a comaximal.## Prove that any two ideal (a) and (b) in PID are comaximal iff gcd(a,b)=1

**Proof:**let R be a PID. We will start with the right implication:

“\Rightarrow“: let d be the generator for the principal ideal generated by a and b, i.e.,

\begin{equation*} (d) = (a,b) = \{ax+by \ | \ x,y\in R\}. \end{equation*}

“\Leftarrow“: reverse the previous proof.