Ring where all elements are zero divisors.

Question

I know that the collection of all nilpotent elements in a commutative ring form such a ring where all elements are zero divisors. For example, if we take $S=\lbrace 0,2,4,6 \rbrace$ as a subring of $\mathbb{Z_8}$, then all elements of $S$ are zero divisors and nilpotent. If we look at $\mathbb{Z_{p^n}}$, then we arrive at another example of this. But I'm struggling to find a well-known example where not every element is nilpotent. More specifically, I want to find such a ring that at least one of the element of the ring is a zero divisor but not nilpotent. How should I think of such an example?

Answer 1

What about $\{0,3,6,9,12,15\}$ where addition and multiplication is $\pmod{18}$?