Let $x$ and $y$ be torsion elements in an R-module $M$ having orders $a$ and $b$. With $a,b \in R$, $a$ and $b$ are relatively prime, and $R$ is a PID. I want to show that $x + y$ has order $ab$.
So if a and b are relatively prime then there exists some $m,n\in R$ s.t.
But i'm not sure where to go from here. Any help would be great.