Let $R$ be a finite commutative ring with unity. Let $a in R$ and define $C_a = {b in R : ann(a) = ann(b)}$.

I want to know the cardinality of the set $C_a$.

For example,

If $a=0$ then $|C_a| = 1$.

If $a$ is a unit then $|C_a| = |U(R)|$ the set of units of $R$.

If $a$ is a zero divisor then I don’t now the answer.

If $R$ is the ring of integers modulo $n$ and $a$ is a non-zero zero divisor then $C_a = {x in mathbb Z_n : (x,n)=d}$ where $a = md$ for a proper divisor $d$ of $n$.

If $R$ is a reduced ring, then $R cong F_{q_1} times cdots times F_{q_k}$, product of finite fields. Define supp(a) = {1 le i le k : a_i is a unit} where $a = (a_1,dots,a_k) in R$. Then $C_a = {b in R: text{supp} (b) = text{supp} (a)}$. From this cardinality of $C_a$ can be obtained.

Thank you.