Mathematics, Department of


Date of this Version

July 2008


A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy. Major: Mathematics. Under the Supervision of Professor Thomas Marley.
Lincoln, Nebraska : August, 2008
Copyright (c) 2008 Livia M. Miller


In Noetherian rings there is a hierarchy among regular, Gorenstein and Cohen-Macaulay rings. Regular non-Noetherian rings were originally defined by Bertin in 1971. In 2007, Hamilton and Marley used Cech cohomology to introduce a theory of Cohen-Macaulay for non-Noetherian rings, answering a question posed by Glaz. This dissertation provides a theory of non-Noetherian Gorenstein rings agreeing with the Noetherian definition, and for which regular rings are Gorenstein, and coherent Gorenstein rings are Cohen-Macaulay. The relationship between Gorenstein rings and FP-injective dimension as defined by Stenstrom is also explored. Finally, an additional characterization of Gorenstein rings involving homological dimensions is examined in the non-Noetherian case.