$\DeclareMathOperator\ann{ann}$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.



Annihilator of an element in a ring
If you allow some $a\_i$ to be $0$, then the answer is obviously yes.


If there are no $0$-divisors in $R$, then the answer is vacuously yes. If there is a $0$-divisor $a$ in $R$, then taking $b = 1$ gives an example where no such sequence exists.


Suppose that $a$ and $b$ both have non-$0$ annihilators. Say $x \ne 0 \ne y$ satisfy $a x = 0$ and $b y = 0$. If $x y \ne 0$, then you may take $a\_1 = x y$. Otherwise, take $a\_1 = x$ and $a\_2 = y$.


