| In the late 1960's Auslander and Bridger published Stable Module Theory, in which the idea of totally reflexive modules first appeared. These modules have been studied by many. However, a bulk of the information known about them is when they are over a Gorenstein ring, since in that case they are exactly the maximal Cohen-Macaulay modules. Much is already known about maximal Cohen-Macaulay modules, that is, totally reflexive modules over a Gorenstein ring. Therefore, we investigate the existence and abundance of totally reflexive modules over non-Gorenstein rings.;It is known that if there exist one non-trivial totally reflexive module over a non- Gorenstein ring, then there exists infinitely many non-trivial non-isomorphic indecomposable ones. Many different techniques are utilized to study the representation theory of this wild category of totally reflexive modules over non-Gorenstein rings, including the classic approach of Auslander-Reiten theory. We present several of these results and conclude by giving a complete description of the totally reflexive modules over a specific family of non-Gorenstein rings. |
