Let $R$ be a commutative ring with unit which is also a PID. Let $M$ be an $R$-module. For $m\in M$, we define $$Ann(m)=\{r\in R:r\cdot m=0\}.$$ Also, if $Ann(m)=\langle c \rangle$ for some $c\in R\backslash \{0\}$, $c$ is called the order of the element $m$.
I want to prove the following:
Suppose $x_1,x_2\in M$ are such that $Ann(x_1)=\langle c_1 \rangle$ and $Ann(x_2)=\langle c_2 \rangle$ for $c_1,c_2\in R\backslash \{0\}$. Then there exists $x\in M$ such that $Ann(x)=\langle lcm(c_1,c_2) \rangle$. ($lcm$ denotes the least common multiple).
Any ideas on how to approach this? I thought this would not be hard to prove but I am quite stuck. Thanks in advance for your help.