If the intersection of all non-zero ideals of an integral domain R is non-zero, show that R is a field. Let $R$ be a commutative ring with unity.  Call $R$ a *-ring if the intersection of all non-zero ideals of $R$ is non-zero.  Note that we consider $R$ itself an ideal.  If $R$ is an integral domain that is a *-ring, show that $R$ is a field.
My idea was to show that a maximal ideal $I$ is the zero ideal, but I'm not really getting anywhere.  Doesn't seem like too hard of a problem, so any hints would be amazing.

Hint: Suppose that the intersection of the principal non-zero ideals is non-zero, i.e. $$0\not= a \in \bigcap_{0 \not=b \in R} bR$$  Note that $a$ is also in the principal ideal $a^2R$.  Use the fact that $R$ is a domain to conclude that $a$ is a unit.  
