Beside the usual rules for non-unital commutative ring (that is, a ring without multiplicative identity) $R$, I want $R$ to satisfy the following:
$ab = 0$ for each element $a$ and there are infinitely many elements $b$ in the ring that satisfies $ab = 0$. Also for each possible $b$, let us label it as $b_1, b_2, ...$ and so on. Then $b_1 b_2 = b_1 b_3 = ... = b_2 b_3 = b_2 b_4 = ... = b_3 b_4 =....= 0$. For each $b_1$, there does not exist $c$ that satisfies $b_1 c = 0$ other than $a$ and $b_i$'s. There exists infinitely many such $a$'s. Also, for every such $a$, there exists infitely many $c$ such that $ac \neq 0$.
Does such ring exist? If so, can ring operation $(+,\cdot)$ can be represented as arithmetic operation?
By the way, the trivial case where every $ab = 0$ is excluded.