  Let R be a noetherian ring which is a finite module over its centre Z(R).
This paper studies the consequences for R of the hypothesis that it is a
maximal Cohen Macaulay Z(R)-module. Old results are reviewed and a number of
new results are proved. The additional hypothesis of homological grade symmetry
is proposed as the appropriate extra lever needed to extend the classical
commutative homological hierarchy to this setting, and results are given
offering evidence in support of this proposal.
