Solutions of an equation in a PID

[ihggin]

Let $$R$$ be a Principal Ideal Domain. Let $$a,b \in R$$, at least one of which is non-zero. Determine precisely all solutions $$(s,t)$$, where $$s,t \in R$$, of the equation $$sa+tb= \gcd (a,b)$$.

My attempt: since $$\gcd (a,b)$$ divides both $$a$$ and $$b$$, we can divide both sides by $$\gcd (a,b)$$ to get the equation $$s \alpha + t \beta = 1$$, where $$\gcd (\alpha , \beta)=1$$. Just playing around in the integers, if we find a solution $$x,y$$, we can generate other solutions by taking $$x+n \beta, y-n \alpha$$. However, I'm unsure of a couple of things: 1) how do you show that this gives all solutions (if it does), and 2) how do you find an initial solutions $$x,y$$? Of course, any other approaches would be nice to see as well.

Thanks.

 

[mathwonk]

can you prove it over Z? does the same proof work?