Let $a$ and $b$ be two elements of a commutative ring $R$ with 1 such that $ann(a) \ne ann(b)$.

is it always possible to find a sequence of elements $a_1,\dots,a_k \in R$ such that $a \in ann(a_1), a_1 \in ann(a_2), \dots, a_{k-1} \in ann(a_k),$ and $a_k \in ann(b)$?

Please share your thoughts or some references. Thank you.