  In this paper we study endomorphism rings of finite global dimension over not
necessarily normal commutative rings. These objects have recently attracted
attention as noncommutative (crepant) resolutions, or NC(C)Rs, of
singularities. We propose a notion of a NCCR over any commutative ring that
appears weaker but subsumes all previous notions. Our results yield strong
necessary and sufficient conditions for the existence of such objects in many
cases of interest. We also give new examples of NCRs of curve singularities,
regular local rings and normal crossing singularities. Moreover, we introduce
and study the global spectrum of a ring $R$, that is, the set of all possible
finite global dimensions of endomorphism rings of MCM $R$-modules. Finally, we
use a variety of methods to compute global dimension for many endomorphism
rings.
