TITLE: An example of a Ring with many zero divisors
QUESTION [14 upvotes]: Is there an example of a commutative ring $R$ with identity such that all its elements distinct from $1$ are zero-divisors?
I know that in a finite ring all the elements are units or zero-divisors. Is there a finite ring with the property I've required?
Obviosuly I'm requiring that $|R|\geq 3$.

REPLY [2 votes]: Yet another ring isomorphic to Gregory's answer which I find worth mentioning: $$(\mathcal{P}(\mathbb{Z}), \Delta, \cap),$$ where $\Delta$ is symmetric difference. Not surprisingly, $\mathbb{Z}$ can be replaced with any set.