$\DeclareMathOperator\ann{ann}$Let $a$ and $b$ be two non-zero zero divisors of a commutative ring $R$ with 1 such that $\ann(a) \ne \ann(b)$.

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

Please share your thoughts or some references.