Research: Presentations of Rings With Non-trivial Self-orthogonal Modules

A result of Foxby, Reiten and Sharp says that a commutative noetherian local ring R admits a dualizing module if and only if R is Cohen—Macaulay and a homomorphic image of a local Gorenstein ring Q. We establish an analogous result by showing that such a ring R having a dualizing module admits a non-trivial finitely generated self-orthogonal module C satisfying Hom_R(C,C) \cong R if and only if R is the homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a non-trivial way, forcing significant structural requirements on the ring R.