Non-commutative resolutions for Segre products and Cohen-Macaulay rings of hereditary representation type

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.