#### Date of Award

5-2017

#### Document Type

Open Access Dissertation

#### Department

Mathematics

#### First Advisor

Adela Vraciu

#### Abstract

For a standard graded Cohen-Macaulay ring R, if the quotient R/(x) admits nonfree totally reflexive modules, where x is a system of parameters consisting of elements of degree one, then so does the ring R. A non-constructive proof of this statement was given in [16]. We give an explicit construction of the totally reflexive modules over R obtained from those over R/(x).

We consider the question of which Stanley-Reisner rings of graphs admit nonfree totally reflexive modules and discuss some examples. For an Artinian local ring (R,m) with m3 = 0 and containing the complex numbers, we describe an explicit construction of uncountably many non-isomorphic indecomposable totally reflexive modules, under the assumption that at least one such non-free module exists. In addition, we generalize Rangel-Tracy rings. We prove that her results do not generalize. Specifically, the presentation of a totally reflexive module cannot be choosen generically in our generalizations.

#### Recommended Citation

Atkins, J.(2017). *On the Existence of Non-Free Totally Reflexive Modules.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/4051