  We study commutative Cohen-Macaulay rings whose Cohen-Macaulay representation
theory are controlled by representations of quivers, which we call hereditary
representation type. Based on tilting theory and cluster tilting theory, we
construct some commutative Cohen-Macaulay rings of hereditary representation
type. First we give a general existence theorem of cluster tilting module or
non-commutative crepant resolutions on the Segre product of two commutative
Gorenstein rings whenever each factor has such an object. As an application we
obtain three examples of Gorenstein rings of hereditary representation type
coming from Segre products of polynomial rings. Next we introduce extended
numerical semigroup rings which generalize numerical semigroup rings and form a
class of one-dimensional Cohen-Macaulay non-domains, and among them we provide
one family of Gorenstein rings of hereditary representation type. Furthermore,
we discuss a $4$-dimensional non-Gorenstein Cohen-Macaulay ring whose
representations are still controlled by a finite dimensional hereditary
algebra. We show that it has a unique $2$-cluster tilting object, and give a
complete classification of rigid Cohen-Macaulay modules, which turns out to be
only finitely many.
