  In this paper we study orders over Cohen-Macaulay rings. We discuss desirable
properties for these orders if they are to represent NCCRs of the base rings.
While some definitions have been made, we discuss an alternate definition and
the non-existence of examples. We then give necessary and sufficient conditions
for an order to have certain desirable homological properties. We examine
examples of rings satisfying these properties to prove that certain
endormorphism rings over abelian quotient singularities have infinite global
dimension.
