Let $R$ be a commutative ring with identity and with the following properties:
(a) The intersection of all of its nonzero ideals is nontrivial.
(b) If $x$ and $y$ are zero divisors in $R$, then $xy=0$.
Prove that $R$ has exactly one nontrivial ideal.
I was trying to prove that $I$, the intersection of all ideals is a maximal ideal. I can see that $I^2=0$ but cannot go any further. I'd much prefer hints than complete answers. Thank you.