  We first generalize classical Auslander-Reiten duality for isolated
singularities to cover singularities with a one-dimensional singular locus. We
then define the notion of CT modules for non-isolated singularities and we show
that these are intimately related to noncommutative crepant resolutions
(NCCRs). When R has isolated singularities, CT modules recover the classical
notion of cluster tilting modules but in general the two concepts differ. Then,
wanting to generalize the notion of NCCRs to cover partial resolutions of Spec
R, in the main body of this paper we introduce a theory of modifying and
maximal modifying modules. Under mild assumptions all the corresponding
endomorphism algebras of the maximal modifying modules for three-dimensional
Gorenstein rings are shown to be derived equivalent. We then develop a theory
of mutation for modifying modules which is similar but different to mutations
arising in cluster tilting theory. Our mutation works in arbitrary dimension,
and in dimension three the behavior of our mutation strongly depends on whether
a certain factor algebra is artinian.
