# ra.rings and algebras – Elements with equal annihilators

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.