How many elements in a ring can be invertible? 
If $R$ is a finite ring (with identity) but not a field, let $U(R)$ be its group of units. Is $\frac{|U(R)|}{|R|}$ bounded away from $1$ over all such rings?

It's been a while since I cracked an algebra book (well, other than trying to solve this recently), so if someone can answer this, I'd prefer not to stray too far from first principles within reason.

By the way, the number $\# U(R)/ \#R$ equals $\sum_{\mathfrak{m}} \left(1-\frac{1}{\# R/\mathfrak{m}}\right)$, where the sum ranges over all maximal ideals $\mathfrak{m} \subseteq R$.
